################################################################################################################################
# Fibers of maximal Naruki cones under the Yoshida matrix and true valuation of Cross_37 on the relative interior of (aa2a3a4) #
################################################################################################################################

# By construction, the Yoshida matrix is not injective on the Bergman fan. Our goal is to analyze the fibers over the relative interiors of the two maximal cone representatives. Given any  maximal cone in the Bergman fan of E6, compute the intersection of its image (after projecting away from the all-ones vector) with the corresponding maximal Naruki cone, and record only those intersecting the relative interior. There are 66 cones in the Bergman fan satisfying this condition with respect to each representative cone (aa2a3a4) and (aa2a3b).

# We do by following four steps:
# 0) compute C = intersection of the image of the Bergman cone with the maximal Naruki cone
# 1) check if C is non-empty and it is not 0-dimensional. If so discard it.
# 2) If C is non trivial, compute the center of mass of the cone C.
# 3) checking if this center of mass lies in the relative interior of the maximal Naruki cone. If the answer is yes, then include the original Bergman cone in the list of cones to consider for determining the fibers. If the answer is no. Discard the cone.

# We save the output for the (aa2a3a4) cone as the dictionary 'aa2a3a4_lower_dim_overlaps' in the file 'Input/aa2a3a4_lower_dim_overlaps.sobj'. Similarly, the data for aa2a3b is recorded as the dictionary 'aa2a3b_lower_dim_overlaps' in the file 'Input/aa2a3b_lower_dim_overlaps.sobj'. 

# Once the search is reduced to a union the cones in 'aa2a3a4_lower_dim_overlaps' and 'aa2a3b_lower_dim_overlaps', we must determine the points on each cone whose image lies in the relative interior of one of the two maximal Naruki cones. For this, we must find which positive linear combinations of the five rays of each cone map to the relative interior of the corresponding Naruki cones. Since the Naruki cones are simplicial, the answer for each ray will be unique if we project away from the all-ones vector, although the scalars might be negative. The solutions are recorded in the dictionaries 'allSolns_aa2a3a4_lower_dim' and 'allSolns_aa2a3b_lower_dim'.

# Once the solutions for each generating ray are determined, we take positive linear combinations (with scalars z1,...,z5) of these solutions and require these expressions to be non-negative. The expressions are stored as 'coefficientsImagesInaa2a3a4_lower_dim' and 'coefficientsImagesInaa2a3b_lower_dim', respectively. In both cases, we confirm that each maximal Naruki cone is contained in the image of the corresponding relevant Bergman cone.

# These inequalities together with the non-negative condition on the variables z1,...,z5 are recorded in the dictionaries 'IneqsFibersInaa2a3a4_lower_dim' and 'IneqsFibersInaa2a3b_lower_dim'. They allow us to build a polyhedron in R^5, with 5, 6 or 7 extremal rays. In turn, these extremal rays determine the rays of the cone computing the fiber of the relative interior of (aa2a3a4) and (aa2a3b), respectively, in the corresponding Bergman cone in 'aa2a3a4_lower_dim_overlaps' and 'aa2a3b_lower_dim_overlaps'. The rays for each of these cones in R^36 are stored in the dictionaries 'newraysFibersInaa2a3a4_lower_dim' and 'newraysFibersInaa2a3ab_lower_dim', respectively.



# The dictionary 'aa2a3a4_lower_dim_overlaps' is used to compute the valuation of the Cross function Cross37 on the relative interior of (aa2a3a4) and confirm that this value always agrees with the expected one. We do so by explicitly identifying the leading terms of roots in Y34 with those in Y38. The identification varies from cone to cone, but some reductions reduce the ammount of computations that need to be performed from 66 to 32.




#########################################
# Data associated to the root system E6 #
#########################################


RootSystemE6 = load("../../General/L6.sobj")

L6 = RootSystemE6.root_lattice()

# L6.dynkin_diagram()
#         O 2
#         |
#         |
# O---O---O---O---O
# 1   3   4   5   6
# E6
#diagram = [{1,3}, {3,4}, {4,2}, {4,5}, {5,6}]

# Weyl group
# These are in the alpha basis

weyl_gens = L6.weyl_group().gens()


os.chdir("../../Yoshida/Scripts")
load("Yoshida.sage")



###############################################
# Data associated to the Bergman fan Berg(E6) #
###############################################

# This data was computed in the folder BergmanFan and in Yoshida.

E6_1_Numbers =  load("../../BergmanFan/Scripts/Input/E6_1_Numbers.sobj")
E6_2_Numbers =  load("../../BergmanFan/Scripts/Input/E6_2_Numbers.sobj")
E6_4_Numbers =  load("../../BergmanFan/Scripts/Input/E6_4_Numbers.sobj")
E6_7_Numbers =  load("../../BergmanFan/Scripts/Input/E6_7_Numbers.sobj")
E6_8_Numbers =  load("../../BergmanFan/Scripts/Input/E6_8_Numbers.sobj")
E6_12_Numbers =  load("../../BergmanFan/Scripts/Input/E6_12_Numbers.sobj")
E6_13_Numbers =  load("../../BergmanFan/Scripts/Input/E6_13_Numbers.sobj")

vertexIndexListE6 = E6_1_Numbers[1] + E6_2_Numbers[1]+E6_4_Numbers[1]+E6_7_Numbers[1]+E6_8_Numbers[1]+E6_12_Numbers[1]+E6_13_Numbers[1]

# We load the Bergman fan of E6:
BergmanFan =  load("../../BergmanFan/Output/BergmanFanOfE6TruncatedFVectorAndOrbitsPerDimension.sobj")




################################################################################
# Functions for computing Bergman Cones and their images under the Yoshida map #
################################################################################


#####################################
# Construction of cones of Berg(E6) #
#####################################

# The following function converts a flat of a matroid given as a complete list of all indices of elements in the ground set contained in the flat.

def flat_to_vertex(flat):
    v = zero_vector(36)
    for root in flat:
        v[root] = 1
    return v

# The following function picks a cell in the Bergman complex (given as a list of vertex indices)  and returns the list of 0-1 incidence vectors giving the rays of the cone over the input cell. 
# vertexIndexListE6 is the list of all flats of the reflection arrangement E6.

# "flat_numbers" = list of vertex indices generating a cell in the Bergman complex of E6

def cone_from_flat_numbers(flat_numbers):
    vs = [flat_to_vertex(vertexIndexListE6[f]) for f in flat_numbers]
    return vs



##########################################################################
# Projection to canonical representatives in a tropical projective torus #
##########################################################################

# The following function picks a vertex and computes its canonical representative in the tropical projective torus by setting the last coordinate to be 0.

# v = vector with real entries.
def projection(v):
    last = v[-1]
    subtract = vector([v[-1] for i in range(0, len(v))])
    return v - subtract





###########################################################################################################
# Finding all Bergman cones in the fiber of the maximal Naruki cone representatives under the Yoshida map #
###########################################################################################################

# We compute the images of all the 142560 maximal cones in the Bergman fan under the Yoshida map.

# allMaxCones0 = []
# for x in BergmanFan[1][4]:
#     for flats in x:
#     	print flats
#     	Ycone=[Yoshida_matrix*v for v in cone_from_flat_numbers(flats)]        
#     	allMaxCones0.append((flats, span(Ycone).dimension()))

#  save(allMaxCones0,'Input/allImagesOfMaxConesBE6.sobj')

allMaxCones0 = load('Input/allImagesOfMaxConesBE6.sobj')

# len(allMaxCones0)
# 142560


# We want to keep only the maximal cones that overlap with the two maximal cone representatives. We start by loading the two maximal cone representatives computed in 'cellsNarukiFan.sage':

aa2a3a4 = load("../../ComputeLines/Scripts/Input/aa2a3a4Cell.sobj")
aa2a3b =  load("../../ComputeLines/Scripts/Input/aa2a3bCell.sobj")

# The following vector records the all-zero vector in R^40:
origin = [projection(Yoshida_matrix*v) for v in cone_from_flat_numbers([687])][0]


##########################################################################
# Classifying the Bergman cones by their images under the Yoshida matrix #
##########################################################################

# We want to determine the fiber of the relative interior of the two maximal Naruki cone representatives on each of the relevant maximal cones in the Naruki fan. For this, we must find which positive linear combinations of the five rays of each cone map to the relative interior of the corresponding Naruki cones. Since the Naruki cones are simplicial, the answer for each ray will be unique if we project away from the all-ones vector, although the scalars might be negative. Once the solutions for each generating ray are determined, we take positive linear combinations are require these expressions to be non-negative. The following scripts perform this task.


#####################################################
# Finding relevant maximal cones in the Bergman fan #
#####################################################

# Notice that if the overlap of the image of a maximal cone in the Bergman fan with a maximal Naruki cone is not 5-dimensional cone, it is still possible for the image to meet the relative interior of the maximal Naruki cone. If so, these cones will still contribute to the computation of the fibers of the Yoshida map.

# The answer to this question can be determined by the following steps:
# 0) compute C = intersection of the image of the Bergman cone with the maximal Naruki cone
# 1) check if C is non-empty and it is not 0-dimensional. If so discard it.
# 2) If C is non trivial, compute the center of mass of the cone C.
# 3) checking if this center of mass lies in the relative interior of the maximal Naruki cone. If the answer is yes, then include the original Bergman cone in the list of cones to consider for determining the fibers. If the answer is no. Discard the cone.

# We save the output for the (aa2a3a4) cone as the dictionary 'aa2a3a4_lower_dim_overlaps' in the file 'Input/aa2a3a4_lower_dim_overlaps.sobj'. Similarly, the data for aa2a3b is recorded as the dictionary 'aa2a3b_lower_dim_overlaps' in the file 'Input/aa2a3b_lower_dim_overlaps.sobj'

# The dictionary 'aa2a3a4_lower_dim_overlaps' consists of 66 tuples (C, dim(C)). We will use it to compute the valuation of the Cross function Cross37 on the relative interior of (aa2a3a4) and confirm that it always has the expected valuation.



########################
# Relevant Naruki rays #
########################

# We start by loading the vertices of the Bergman fan:
vBerg=load("../../BergmanFan/Output/CanonicalCoordsOfVerticesBE6.sobj")

# We compute the relevant 4 rays in the Bergman fan yielding the generators of the 5 relevant rays in the Naruki fan.
v0=vector(vBerg[0])
v1=vector(vBerg[1])
v4=vector(vBerg[4])
v158=vector(vBerg[158])
v36=vector(vBerg[36])

va2 = v0 +v1
va3 = v0 + v1 +v4
va4 = v0 + v1 +v158
va = v0
vb = v36

# # We compute their images under the Yoshida matrix and their projection away from the all-ones vector in R^40.

YMa = projection(Yoshida_matrix*vector(va))
YMa2 = projection(Yoshida_matrix*vector(va2))
YMa3 = projection(Yoshida_matrix*vector(va3))
YMa4 = projection(Yoshida_matrix*vector(va4))
YMb = projection(Yoshida_matrix*vector(vb))


# Matrices encoding the rays of the 2 maximal Naruki cone representatives

Maa2a3a4 = matrix([YMa,YMa2,YMa3,YMa4]).transpose()
Maa2a3b = matrix([YMa,YMa2,YMa3,YMb]).transpose()

#############
# TYPE AAAA #
#############

# aa2a3a4_lower_dim_overlaps = dict()
# for c in allMaxCones0:
#     print c
#     P = Polyhedron(rays=[projection(Yoshida_matrix*v) for v in cone_from_flat_numbers(c[0]) if projection(Yoshida_matrix*v)!=origin]).intersection(aa2a3a4)
#     if P.rays() !=():
#        CoMofP = sum([vector(r) for r in P.rays()])
#        try:
# 	soln12 = Maa2a3a4 \ CoMofP
# 	if all([bool(soln12[k] > 0) for k in range(0,4)]) == True:
# 	   print 'found new cone!'
# 	   aa2a3a4_lower_dim_overlaps[tuple(c[0])] = (P,P.dim())
# 	   save(aa2a3a4_lower_dim_overlaps, 'Input/aa2a3a4_lower_dim_overlaps.sobj')	   
#        except ValueError:
#     	   print 'no solution'
#     else:
# 	print 'empty polyhedron!'

 aa2a3a4_lower_dim_overlaps = load('Input/aa2a3a4_lower_dim_overlaps.sobj')	  

len(aa2a3a4_lower_dim_overlaps.keys())
66
print aa2a3a4_lower_dim_overlaps.keys()
[(4, 29, 309, 613, 732), (0, 1, 4, 204, 717), (1, 29, 238, 563, 732), (1, 4, 29, 309, 732), (1, 29, 238, 548, 732), (0, 4, 172, 506, 722), (0, 1, 29, 238, 716), (0, 1, 29, 165, 493), (0, 4, 172, 487, 722), (0, 4, 172, 506, 700), (0, 1, 4, 172, 506), (0, 1, 4, 158, 473), (0, 4, 29, 309, 613), (0, 1, 158, 473, 716), (0, 1, 158, 482, 716), (0, 4, 29, 309, 722), (1, 4, 204, 540, 732), (4, 29, 309, 613, 712), (0, 1, 4, 172, 717), (1, 29, 238, 563, 709), (0, 1, 29, 158, 716), (0, 1, 29, 238, 563), (0, 1, 4, 158, 717), (0, 4, 29, 165, 722), (0, 1, 4, 172, 692), (1, 29, 238, 548, 704), (0, 4, 29, 309, 691), (0, 29, 165, 493, 699), (0, 1, 29, 165, 490), (0, 29, 165, 490, 722), (1, 4, 204, 543, 703), (0, 4, 29, 309, 636), (0, 29, 165, 490, 695), (4, 29, 309, 636, 713), (0, 1, 4, 204, 543), (0, 1, 4, 158, 482), (1, 4, 29, 238, 732), (0, 29, 165, 490, 698), (1, 29, 238, 563, 705), (0, 29, 165, 493, 722), (0, 4, 172, 506, 701), (0, 4, 172, 487, 694), (1, 29, 238, 548, 708), (0, 4, 29, 172, 722), (1, 4, 204, 543, 732), (4, 29, 309, 613, 710), (1, 4, 204, 543, 707), (0, 1, 158, 482, 690), (0, 1, 4, 172, 487), (0, 1, 4, 158, 691), (4, 29, 309, 636, 711), (0, 1, 29, 238, 692), (0, 29, 165, 493, 696), (0, 4, 172, 487, 697), (0, 1, 29, 165, 687), (0, 1, 4, 204, 540), (1, 4, 29, 204, 732), (1, 4, 204, 540, 702), (0, 1, 158, 482, 693), (0, 1, 29, 238, 548), (0, 1, 29, 165, 716), (0, 1, 4, 204, 687), (4, 29, 309, 636, 732), (1, 4, 204, 540, 706), (0, 1, 158, 473, 689), (0, 1, 158, 473, 688)]


# We record this information as the tuple (rays of P, dim(P)):

# print {c: (aa2a3a4_lower_dim_overlaps[c][0].rays(), aa2a3a4_lower_dim_overlaps[c][1]) for c in aa2a3a4_lower_dim_overlaps.keys()}
# {(4, 29, 309, 613, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 4, 204, 717): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 29, 238, 563, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 29, 309, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 29, 238, 548, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 238, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 165, 493): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 4, 172, 487, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 172, 506, 700): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 172, 506): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 158, 473): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 29, 309, 613): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 158, 473, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 158, 482, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 29, 309, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 204, 540, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 172, 717): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 29, 238, 563, 709): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 158, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 238, 563): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 4, 158, 717): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 4, 29, 165, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 4, 158, 691): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 172, 692): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 29, 309, 636): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 4, 204, 687): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 29, 309, 691): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 165, 490): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 29, 238, 563, 705): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 204, 543, 703): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 29, 165, 493, 699): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 29, 165, 490, 695): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 165, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 29, 165, 493, 696): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 4, 204, 543): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 158, 482): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 158, 473, 689): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 172, 487, 697): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 4, 172, 506, 701): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 29, 165, 490, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 29, 165, 493, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (4, 29, 309, 613, 712): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 4, 172, 487, 694): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (1, 29, 238, 548, 708): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 4, 29, 172, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 204, 543, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 29, 165, 687): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 204, 543, 707): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 158, 482, 690): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 172, 487): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (4, 29, 309, 613, 710): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (4, 29, 309, 636, 711): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 1, 29, 238, 692): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (4, 29, 309, 636, 713): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 29, 165, 490, 698): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (0, 4, 172, 506, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 4, 204, 540): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (1, 4, 29, 204, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 204, 540, 702): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 158, 482, 693): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 29, 238, 548): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 29, 238, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 29, 238, 548, 704): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (4, 29, 309, 636, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0)), 4), (1, 4, 204, 540, 706): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4), (0, 1, 158, 473, 688): ((A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (-1, 2, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 2, 0, 0, 0, 2, 0, -1, 0, 2, 0, 0, 0, 2, 0, -1, -1, -1, 2, 0, 2, 2, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0)), 4)}


################################
# From candidate cones to rays #
################################

# Once the search is reduced to a union of few cones, we must determine the points on each cone whose image lies in the relative interior of one of the two maximal Naruki cones. For this, we must find which positive linear combinations of the five rays of each cone map to the relative interior of the corresponding Naruki cones. Since the Naruki cones are simplicial, the answer for each ray will be unique if we project away from the all-ones vector, although the scalars might be negative. Once the solutions for each generating ray are determined, we take positive linear combinations are require these expressions to be non-negative. The following scripts all systems and take their combinations. 


########################################
# Collecting the relevant Bergman rays #
########################################

# We start by recording the list of all Bergman rays appearing in the list of 66 Bergman cones. There is a total of 53 rays to consider.

allRaysaa2a3a4_lower_dimensional = []
for x in aa2a3a4_lower_dim_overlaps.keys():
    allRaysaa2a3a4_lower_dimensional = sorted(list(set(allRaysaa2a3a4_lower_dimensional + list(x))))

len(allRaysaa2a3a4_lower_dimensional)
53

# print allRaysaa2a3a4_lower_dimensional
# [0, 1, 4, 29, 158, 165, 172, 204, 238, 309, 473, 482, 487, 490, 493, 506, 540, 543, 548, 563, 613, 636, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 716, 717, 722, 732]



# We compute the rays of the intersection of the image with our the (aa2a3a4) maximal cone representative in the Naruki fan:
aa2a3a4_fine_overlaps_lower_dim = dict()
for c in aa2a3a4_lower_dim_overlaps.keys():
    print c
    aa2a3a4_fine_overlaps_lower_dim[c] = aa2a3a4_lower_dim_overlaps[c][0].rays()


# # We confirm that the Naruki cone (aa2a3a4) is contained in the image of the corresponding Bergman cones
# [c for c in aa2a3a4_fine_overlaps_lower_dim.keys() if aa2a3a4_lower_dim_overlaps[c][0] != aa2a3a4]
# []


# We check which rays (if any) map to the origin. We find 27 of them do.
vanishingImagesaa2a3a4_lower_dim = [x for x in allRaysaa2a3a4_lower_dimensional if projection(Yoshida_matrix*cone_from_flat_numbers([x])[0])==origin]

# len(vanishingImagesaa2a3a4_lower_dim)
# 27

# print vanishingImagesaa2a3a4_lower_dim
# [687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713]


#############
# TYPE AAAB #
#############

# aa2a3b_lower_dim_overlaps = dict()
# for c in allMaxCones0:
#     print c
#     P = Polyhedron(rays=[projection(Yoshida_matrix*v) for v in cone_from_flat_numbers(c[0]) if projection(Yoshida_matrix*v)!=origin]).intersection(aa2a3b)
#     if P.rays() !=():
#        CoMofP = sum([vector(r) for r in P.rays()])
#        try:
# 	soln12 = Maa2a3b \ CoMofP
# 	if all([bool(soln12[k] > 0) for k in range(0,4)]) == True:
# 	   print 'found new cone!'
# 	   aa2a3b_lower_dim_overlaps[tuple(c[0])] = (P,P.dim())
# 	   save(aa2a3b_lower_dim_overlaps, 'Input/aa2a3b_lower_dim_overlaps.sobj')	   
#        except ValueError:
#     	   print 'no solution'
#     else:
# 	print 'empty polyhedron!'


aa2a3b_lower_dim_overlaps = load('Input/aa2a3b_lower_dim_overlaps.sobj')	  

len(aa2a3b_lower_dim_overlaps.keys())
66

print aa2a3b_lower_dim_overlaps.keys()
[(0, 1, 55, 482, 693), (0, 1, 4, 55, 482), (1, 4, 55, 238, 732), (0, 4, 36, 487, 694), (0, 1, 4, 73, 691), (0, 1, 4, 73, 540), (0, 4, 73, 506, 722), (1, 4, 73, 540, 732), (0, 1, 36, 165, 716), (1, 4, 73, 540, 706), (4, 73, 309, 636, 711), (0, 36, 165, 490, 695), (4, 73, 309, 636, 713), (0, 4, 73, 309, 613), (0, 36, 165, 493, 696), (1, 4, 55, 543, 707), (0, 4, 36, 487, 722), (0, 1, 4, 55, 692), (0, 4, 73, 309, 722), (0, 1, 4, 36, 55), (1, 4, 55, 543, 732), (4, 73, 309, 636, 732), (1, 4, 55, 73, 732), (1, 55, 238, 563, 705), (0, 1, 4, 55, 543), (0, 36, 165, 493, 722), (0, 1, 4, 55, 73), (0, 1, 36, 165, 687), (0, 4, 73, 309, 636), (0, 4, 36, 487, 697), (4, 73, 309, 613, 732), (0, 36, 165, 493, 699), (4, 73, 309, 613, 712), (1, 4, 73, 309, 732), (0, 1, 36, 473, 689), (0, 4, 73, 506, 701), (1, 55, 238, 548, 708), (0, 1, 55, 482, 690), (0, 4, 36, 73, 722), (0, 1, 36, 473, 688), (1, 55, 238, 548, 704), (0, 1, 55, 482, 716), (0, 1, 4, 36, 473), (4, 73, 309, 613, 710), (0, 1, 4, 73, 506), (1, 55, 238, 563, 732), (1, 4, 55, 543, 703), (1, 4, 73, 540, 702), (0, 4, 36, 165, 722), (0, 1, 36, 165, 490), (0, 1, 36, 165, 493), (0, 1, 4, 36, 73), (0, 1, 36, 473, 716), (0, 1, 55, 238, 563), (0, 1, 55, 238, 548), (0, 36, 165, 490, 698), (1, 55, 238, 563, 709), (1, 55, 238, 548, 732), (0, 1, 4, 36, 487), (0, 1, 36, 55, 716), (0, 4, 73, 309, 691), (0, 36, 165, 490, 722), (0, 1, 4, 36, 687), (0, 4, 73, 506, 700), (0, 1, 55, 238, 692), (0, 1, 55, 238, 716)]


# We record this information as the tuple (rays of P, dim(P)):
# print {c: (aa2a3b_lower_dim_overlaps[c][0].rays(), aa2a3b_lower_dim_overlaps[c][1]) for c in aa2a3b_lower_dim_overlaps.keys()}
# {(0, 1, 55, 482, 693): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 55, 482): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 55, 238, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 36, 487, 694): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 73, 691): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 73, 540): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 73, 506, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 55, 238, 563, 705): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 36, 165, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 36, 165, 490, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 73, 540, 706): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (4, 73, 309, 636, 711): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 36, 165, 490, 695): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (4, 73, 309, 636, 713): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 4, 73, 309, 613): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 36, 165, 493, 696): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 55, 543, 707): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 36, 487, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 55, 692): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 73, 309, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 36, 55): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 55, 543, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 55, 238, 548, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 55, 73, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 55, 543): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 55, 543, 703): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 55, 73): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 36, 487): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 36, 165, 687): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 73, 309, 636): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 4, 36, 487, 697): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 36, 165, 493, 699): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 36, 165, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 73, 309, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 36, 687): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 36, 473, 689): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 73, 506, 701): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 55, 238, 548, 708): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 55, 482, 690): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 36, 73, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 36, 473, 688): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 55, 482, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 36, 473): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (4, 73, 309, 613, 710): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (1, 4, 73, 540, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 55, 238, 563, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 4, 36, 73): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 36, 165, 493, 722): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 55, 238, 548, 704): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (4, 73, 309, 613, 712): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 1, 36, 165, 490): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 36, 165, 493): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 4, 73, 540, 702): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 36, 473, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 55, 238, 563): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 55, 238, 548): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 36, 165, 490, 698): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (4, 73, 309, 636, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (4, 73, 309, 613, 732): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 1, 36, 55, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 73, 309, 691): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0)), 4), (0, 1, 4, 73, 506): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (1, 55, 238, 563, 709): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 4, 73, 506, 700): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 55, 238, 692): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4), (0, 1, 55, 238, 716): ((A ray in the direction (0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, 0, 2, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0), A ray in the direction (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0), A ray in the direction (-1, 1, 0, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, 0, -1, 1, -1, 0, 0, 2, 0, -1, 0, 1, -1, -1, -1, 1, 0, -1, -1, -1, 1, 0, 1, 2, 0), A ray in the direction (-1, 0, -1, -1, 0, -1, 0, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, -1, 0, -1, 0, 0, 1, 0, -1, -1, 1, -1, -1, -1, 0, -1, -1, -1, -1, 0, 0, 1, 1, 0)), 4)}



################################
# From candidate cones to rays #
################################

# Once the search is reduced to a union of few cones, we must determine the points on each cone whose image lies in the relative interior of one of the two maximal Naruki cones. For this, we must find which positive linear combinations of the five rays of each cone map to the relative interior of the corresponding Naruki cones. Since the Naruki cones are simplicial, the answer for each ray will be unique if we project away from the all-ones vector, although the scalars might be negative. Once the solutions for each generating ray are determined, we take positive linear combinations are require these expressions to be non-negative. The following scripts all systems and take their combinations. 


########################################
# Collecting the relevant Bergman rays #
########################################

# We start by recording the list of all Bergman rays appearing in the list of 66 Bergman cones. There is a total of 66 rays to consider.

allRaysaa2a3b_lower_dimensional = []
for x in aa2a3b_lower_dim_overlaps.keys():
    allRaysaa2a3b_lower_dimensional = sorted(list(set(allRaysaa2a3b_lower_dimensional + list(x))))

len(allRaysaa2a3b_lower_dimensional)
51

# print allRaysaa2a3b_lower_dimensional
# [0, 1, 4, 36, 55, 73, 165, 238, 309, 473, 482, 487, 490, 493, 506, 540, 543, 548, 563, 613, 636, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 716, 722, 732]
 


# We compute the rays of the intersection of the image with our the (aa2a3b) maximal cone representative in the Naruki fan:
aa2a3b_fine_overlaps_lower_dim = dict()
for c in aa2a3b_lower_dim_overlaps.keys():
    print c
    aa2a3b_fine_overlaps_lower_dim[c] = aa2a3b_lower_dim_overlaps[c][0].rays()


# # We confirm that the Naruki cone (aa2a3b) is contained in the image of the corresponding Bergman cones
# [c for c in aa2a3b_fine_overlaps_lower_dim.keys() if aa2a3b_lower_dim_overlaps[c][0] != aa2a3b]
# []


# We check which rays (if any) map to the origin. We find 27 of them do.
vanishingImagesaa2a3b_lower_dim = [x for x in allRaysaa2a3b_lower_dimensional if projection(Yoshida_matrix*cone_from_flat_numbers([x])[0])==origin]

# len(vanishingImagesaa2a3b_lower_dim)
# 27

# print vanishingImagesaa2a3b_lower_dim
# [687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713]

# We confirm that the rays with vanishing images on each cone type agree
# vanishingImagesaa2a3a4_lower_dim == vanishingImagesaa2a3b_lower_dim
# True



#####################################################################################################################
# Finding the fibers of Yoshida map at the relative interiors of the Naruki cones within each relevant Bergman cone #
#####################################################################################################################


# Our next goal is to determine the fiber of the relative interior of the two maximal Naruki cone representatives on each of the relevant maximal cones in the Bergman fan computed earlier (listed in 'aa2a3a4_lower_dim_overlaps' and 'aa2a3b_lower_overlaps', respectively). For this, we must find which positive linear combinations of the five rays of each cone map to the relative interior of the corresponding Naruki cones. Since the Naruki cones are simplicial, the answer for each ray will be unique if we project away from the all-ones vector, although the scalars might be negative. The solutions are recorded in the dictionaries 'allSolns_aa2a3a4_lower_dim' and 'allSolns_aa2a3b_lower_dim'.

# Once the solutions for each generating ray are determined, we take positive linear combinations (with scalars z1,...,z5) of these solutions and require these expressions to be non-negative. The expressions are stored as 'coefficientsImagesInaa2a3a4_lower_dim' and 'coefficientsImagesInaa2a3b_lower_dim', respectively.
 
# These inequalities together with the non-negative condition on the variables z1,...,z5 are recorded in the dictionaries 'IneqsFibersInaa2a3a4_lower_dim' and 'IneqsFibersInaa2a3b_lower_dim'. They allow us to build a polyhedron in R^5, with 5, 6 or 7 extremal rays. In turn, these extremal rays determine the rays of the cone computing the fiber of the relative interior of (aa2a3a4) and (aa2a3b), respectively, in the corresponding Bergman cone in 'aa2a3a4_lower_dim_overlaps' and 'aa2a3b_lower_dim_overlaps'. The rays for each of these cones in R^36 are stored in the dictionaries 'newraysFibersInaa2a3a4_lower_dim' and 'newraysFibersInaa2a3ab_lower_dim', respectively.


#########################################################################
# Scalars in the Naruki cone of the images of the relevant Bergman rays #
#########################################################################

#############
# TYPE AAAA #
#############

# We collect the images of all relevant rays of the Bergman fan under the Yoshida matrix:
vYM = {k: projection(Yoshida_matrix*vector(vBerg[k])) for k in sorted(list(set(allRaysaa2a3a4_lower_dimensional)))}

# We want to find the solution sets. The solutions will be unique (if they exist) since the matrix Maa2a3a4 has maximal rank (4).

allSolns_aa2a3a4_lower_dim = dict()
for k in allRaysaa2a3a4_lower_dimensional:
    print k
    try:
	soln12 = Maa2a3a4 \ vYM[k] 
	allSolns_aa2a3a4_lower_dim[k] = soln12
    except ValueError:
    	   print 'no solution'

len(allSolns_aa2a3a4_lower_dim.keys())
53

# # We record the information:
# print allSolns_aa2a3a4_lower_dim
# {0: (1, 0, 0, 0), 1: (-1, 1, 0, 0), 4: (0, -1, 1, 0), 716: (0, -3, 3, 0), 691: (0, 0, 0, 0), 540: (2, 0, 0, 0), 29: (0, 0, -1, 1), 158: (0, -1, 0, 1), 543: (2, 0, 0, 0), 548: (2, 0, 0, 0), 165: (-1, 0, 1, 0), 172: (-1, 1, -1, 1), 687: (0, 0, 0, 0), 688: (0, 0, 0, 0), 689: (0, 0, 0, 0), 690: (0, 0, 0, 0), 563: (2, 0, 0, 0), 692: (0, 0, 0, 0), 309: (0, 1, 0, 0), 694: (0, 0, 0, 0), 695: (0, 0, 0, 0), 696: (0, 0, 0, 0), 697: (0, 0, 0, 0), 698: (0, 0, 0, 0), 699: (0, 0, 0, 0), 700: (0, 0, 0, 0), 701: (0, 0, 0, 0), 702: (0, 0, 0, 0), 693: (0, 0, 0, 0), 704: (0, 0, 0, 0), 705: (0, 0, 0, 0), 706: (0, 0, 0, 0), 707: (0, 0, 0, 0), 708: (0, 0, 0, 0), 709: (0, 0, 0, 0), 710: (0, 0, 0, 0), 711: (0, 0, 0, 0), 712: (0, 0, 0, 0), 713: (0, 0, 0, 0), 204: (1, 0, -1, 1), 717: (0, 0, -3, 3), 722: (-3, 3, 0, 0), 473: (0, -2, 2, 0), 732: (3, 0, 0, 0), 482: (0, -2, 2, 0), 613: (2, 0, 0, 0), 487: (-2, 2, 0, 0), 490: (-2, 2, 0, 0), 493: (-2, 2, 0, 0), 238: (1, -1, 1, 0), 506: (-2, 2, 0, 0), 703: (0, 0, 0, 0), 636: (2, 0, 0, 0)}


###############################
# Combining solutions of rays #
###############################

# Next, we write down the images on all relevant cones using the variables z1,...,z5

zs = [var("z%s"%i) for i in range(1,6)]
PolyZ = PolynomialRing(QQ,zs)

coefficientsImagesInaa2a3a4_lower_dim = dict()
for u in aa2a3a4_fine_overlaps_lower_dim.keys():
    coefficientsImagesInaa2a3a4_lower_dim[u] = sum([zs[i]*vector(allSolns_aa2a3a4_lower_dim[u[i]]) for i in range(0,5)])


# We write down dictionaries corresponding to the delta function on each singleton of zs. This will simplify the computation of the linear combinations of zi's that must be non-negative.

dictzs = dict()
for i in range(1,6):
    dictzs[i] = dict()
    for j in range(0,5):
    	if j == i-1:
	   dictzs[i][zs[j]] = 1
	else:
		dictzs[i][zs[j]] = 0

# dictzs
# {1: {z5: 0, z4: 0, z3: 0, z2: 0, z1: 1},
#  2: {z5: 0, z4: 0, z3: 0, z2: 1, z1: 0},
#  3: {z5: 0, z4: 0, z3: 1, z2: 0, z1: 0},
#  4: {z5: 0, z4: 1, z3: 0, z2: 0, z1: 0},
#  5: {z5: 1, z4: 0, z3: 0, z2: 0, z1: 0}}


# We write down the inequalities on the scalars z1,...,z5 arising from the positivity of the coefficients of the scalars on the maximal cone (aa2a3a4)

IneqsFibersInaa2a3a4_lower_dim = dict()
for u in coefficientsImagesInaa2a3a4_lower_dim.keys():
    w = coefficientsImagesInaa2a3a4_lower_dim[u]
    # we start by adding all zs[i] >= 0:
    basisOfSolns = [(0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1)]
    for p in range(0,len(w)):
    	basisOfSolns.append(tuple([0] + [w[p].substitute(dictzs[i]) for i in range(1,6)]))
    IneqsFibersInaa2a3a4_lower_dim[u] = basisOfSolns


# Each list of inequalities satisfied by the variables z1,...,z5 yield a polyhedron in R^5. We create the polytopes and their rays:

newraysFibersInaa2a3a4_lower_dim = dict()
for u in IneqsFibersInaa2a3a4_lower_dim.keys():
    rayScalars = Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[u]).rays()
    newraysFibersInaa2a3a4_lower_dim[u] = [vector(x) for x in Polyhedron(rays = [vector(rayScalars[j])*matrix(cone_from_flat_numbers(list(u))) for j in range(0,len(rayScalars))]).rays()]


# set([len(x) for x in newraysFibersInaa2a3a4_lower_dim.values()])
# {5, 6, 7}
# print newraysFibersInaa2a3a4_lower_dim
{(4, 29, 309, 613, 732): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1)], (0, 1, 4, 204, 717): [(0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (4, 4, 0, 1, 4, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0)], (0, 4, 172, 487, 697): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1), (2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0), (3, 0, 1, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0)], (1, 29, 238, 563, 732): [(0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1), (0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1), (0, 4, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1)], (1, 4, 29, 309, 732): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 4, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 4, 0, 1, 4, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 4, 0, 1, 4, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 1, 1, 1)], (1, 29, 238, 548, 732): [(0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1), (0, 3, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 4, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1)], (0, 4, 172, 506, 722): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (4, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0), (4, 0, 1, 0, 4, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0)], (0, 1, 29, 238, 716): [(0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1), (4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 4, 0, 0, 0, 0, 1, 1)], (0, 1, 29, 165, 493): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0), (2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0), (3, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0)], (0, 4, 172, 487, 722): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0), (3, 0, 1, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0), (4, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0), (4, 0, 1, 0, 4, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0)], (0, 4, 172, 506, 700): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1), (2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0)], (0, 1, 4, 172, 506): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0)], (0, 1, 4, 158, 473): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 3, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 1, 158, 482, 693): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1), (2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1)], (0, 1, 29, 238, 548): [(0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 1), (0, 3, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 1, 29, 165, 716): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0), (2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0), (4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1), (4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 4, 0, 0, 0, 0, 1, 1)], (0, 1, 4, 204, 687): [(0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0)], (4, 29, 309, 636, 732): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1)], (1, 4, 204, 540, 706): [(0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0), (0, 2, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 3, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 1, 158, 473, 688): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0), (2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (3, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)]}


#############
# TYPE AAAB #
#############

# We collect the images of all relevant rays of the Bergman fan under the Yoshida matrix:
vYMb = {k: projection(Yoshida_matrix*vector(vBerg[k])) for k in sorted(list(set(allRaysaa2a3b_lower_dimensional)))}

# We want to find the solution sets. The solutions will be unique (if they exist) since the matrix Maa2a3a4 has maximal rank (4).

allSolns_aa2a3b_lower_dim = dict()
for k in allRaysaa2a3b_lower_dimensional:
    print k
    try:
	soln12 = Maa2a3b \ vYMb[k] 
	allSolns_aa2a3b_lower_dim[k] = soln12
    except ValueError:
    	   print 'no solution'

len(allSolns_aa2a3b_lower_dim.keys())
52

# # We record the information:
# print allSolns_aa2a3b_lower_dim
# {0: (1, 0, 0, 0), 1: (-1, 1, 0, 0), 4: (0, -1, 1, 0), 691: (0, 0, 0, 0), 540: (2, 0, 0, 0), 543: (2, 0, 0, 0), 548: (2, 0, 0, 0), 165: (-1, 0, 1, 0), 713: (0, 0, 0, 0), 687: (0, 0, 0, 0), 688: (0, 0, 0, 0), 689: (0, 0, 0, 0), 690: (0, 0, 0, 0), 563: (2, 0, 0, 0), 692: (0, 0, 0, 0), 309: (0, 1, 0, 0), 694: (0, 0, 0, 0), 55: (0, 0, 0, 1), 696: (0, 0, 0, 0), 697: (0, 0, 0, 0), 698: (0, 0, 0, 0), 699: (0, 0, 0, 0), 700: (0, 0, 0, 0), 701: (0, 0, 0, 0), 702: (0, 0, 0, 0), 36: (0, 0, 0, 1), 704: (0, 0, 0, 0), 705: (0, 0, 0, 0), 706: (0, 0, 0, 0), 707: (0, 0, 0, 0), 708: (0, 0, 0, 0), 709: (0, 0, 0, 0), 710: (0, 0, 0, 0), 711: (0, 0, 0, 0), 712: (0, 0, 0, 0), 73: (0, 0, 0, 1), 695: (0, 0, 0, 0), 716: (0, -3, 3, 0), 693: (0, 0, 0, 0), 722: (-3, 3, 0, 0), 473: (0, -2, 2, 0), 732: (3, 0, 0, 0), 482: (0, -2, 2, 0), 613: (2, 0, 0, 0), 487: (-2, 2, 0, 0), 490: (-2, 2, 0, 0), 493: (-2, 2, 0, 0), 238: (1, -1, 1, 0), 506: (-2, 2, 0, 0), 703: (0, 0, 0, 0), 636: (2, 0, 0, 0)}


###############################
# Combining solutions of rays #
###############################

# Next, we write down the images on all relevant cones using the variables z1,...,z5

zs = [var("z%s"%i) for i in range(1,6)]
PolyZ = PolynomialRing(QQ,zs)

coefficientsImagesInaa2a3b_lower_dim = dict()
for u in aa2a3b_fine_overlaps_lower_dim.keys():
    coefficientsImagesInaa2a3b_lower_dim[u] = sum([zs[i]*vector(allSolns_aa2a3b_lower_dim[u[i]]) for i in range(0,5)])


# We write down dictionaries corresponding to the delta function on each singleton of zs. This will simplify the computation of the linear combinations of zi's that must be non-negative.

dictzs = dict()
for i in range(1,6):
    dictzs[i] = dict()
    for j in range(0,5):
    	if j == i-1:
	   dictzs[i][zs[j]] = 1
	else:
		dictzs[i][zs[j]] = 0

# dictzs
# {1: {z5: 0, z4: 0, z3: 0, z2: 0, z1: 1},
#  2: {z5: 0, z4: 0, z3: 0, z2: 1, z1: 0},
#  3: {z5: 0, z4: 0, z3: 1, z2: 0, z1: 0},
#  4: {z5: 0, z4: 1, z3: 0, z2: 0, z1: 0},
#  5: {z5: 1, z4: 0, z3: 0, z2: 0, z1: 0}}


# We write down the inequalities on the scalars z1,...,z5 arising from the positivity of the coefficients of the scalars on the maximal cone (aa2a3b):

IneqsFibersInaa2a3b_lower_dim = dict()
for u in coefficientsImagesInaa2a3b_lower_dim.keys():
    w = coefficientsImagesInaa2a3b_lower_dim[u]
    # we start by adding all zs[i] >= 0:
    basisOfSolns = [(0,1,0,0,0,0), (0,0,1,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1)]
    for p in range(0,len(w)):
    	basisOfSolns.append(tuple([0] + [w[p].substitute(dictzs[i]) for i in range(1,6)]))
    IneqsFibersInaa2a3b_lower_dim[u] = basisOfSolns


# Each list of inequalities satisfied by the variables z1,...,z5 yield a polyhedron in R^5. We create the polytopes and their rays:

newraysFibersInaa2a3b_lower_dim = dict()
for u in IneqsFibersInaa2a3b_lower_dim.keys():
    rayScalars = Polyhedron(ieqs=IneqsFibersInaa2a3b_lower_dim[u]).rays()
    newraysFibersInaa2a3b_lower_dim[u] = [vector(x) for x in Polyhedron(rays = [vector(rayScalars[j])*matrix(cone_from_flat_numbers(list(u))) for j in range(0,len(rayScalars))]).rays()]


# set([len(x) for x in newraysFibersInaa2a3b_lower_dim.values()])
# {5, 6, 7}
# print newraysFibersInaa2a3b_lower_dim
# {(0, 1, 55, 482, 693): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1), (3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1)], (0, 1, 4, 55, 482): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1)], (1, 4, 55, 238, 732): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 4, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 4, 0, 1, 4, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1)], (0, 4, 73, 309, 613): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 36, 165, 490, 722): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0), (3, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0), (4, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0)], (0, 1, 4, 73, 540): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 3, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 4, 73, 506, 722): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (4, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0), (4, 0, 1, 0, 4, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0)], (1, 4, 73, 540, 732): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 3, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 3, 0, 1, 3, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 4, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 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0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 1, 55, 238, 548): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 3, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)], (0, 36, 165, 490, 698): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0), (2, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0), (3, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0)], (1, 55, 238, 548, 732): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 3, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 4, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1)], (0, 1, 4, 36, 487): [(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (3, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0), (3, 0, 1, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0)], (0, 1, 36, 55, 716): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1)], (0, 4, 73, 309, 691): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0)], (0, 1, 4, 73, 506): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0)], (1, 55, 238, 563, 709): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1), (0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1)], (0, 4, 73, 506, 700): [(0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1), (3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0), (3, 0, 0, 0, 3, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0)], (0, 1, 55, 238, 692): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1)], (0, 1, 55, 238, 716): [(0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1), (0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1), (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1)]}





################################################################################################
# Computing the expected and true valuations for Cross37 on the relative interior of (aa2a3a4) #
################################################################################################

# The following function takes a list of lists, and returns a list of all the elements (without repetitions)

def combineLists(listOfList):
    newList = []
    for x in listOfList:
    	for y in x:
    	    newList.append(y)
    return list(set(newList))

#########################################
# Finding relevant Roots for each fiber #
#########################################

# We collect the positive coordinates for each of the rays:

newrootsForConesaa2a3a4 = dict()
for u in newraysFibersInaa2a3a4_lower_dim.keys():
    newrootsForConesaa2a3a4[u] = sorted(combineLists([[k for k in range(0,36) if x[k]!=0] for x in newraysFibersInaa2a3a4_lower_dim[u]]))

# We record the data:
for u in newrootsForConesaa2a3a4.keys():
    print str(u) + ': ' + str(newrootsForConesaa2a3a4[u])
# 
# (4, 29, 309, 613, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (0, 1, 4, 204, 717): [0, 1, 3, 4, 7, 10, 13, 18, 23, 24, 25, 27, 28, 30, 31]
# (1, 29, 238, 563, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (1, 4, 29, 309, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (1, 29, 238, 548, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (0, 1, 29, 238, 716): [0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35]
# (0, 1, 29, 165, 493): [0, 1, 2, 8, 9, 21, 26, 28, 29, 31, 33]
# (0, 4, 172, 487, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (0, 4, 172, 506, 700): [0, 3, 4, 5, 8, 10, 14, 15, 16, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35]
# (0, 1, 4, 172, 506): [0, 1, 4, 5, 8, 21, 26, 27, 28, 30, 31]
# (0, 1, 4, 158, 473): [0, 1, 2, 4, 9, 14, 17, 18, 23, 24, 25]
# (0, 4, 29, 309, 613): [0, 3, 4, 5, 8, 10, 14, 29, 32, 33, 34]
# (0, 1, 158, 473, 716): [0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35]
# (0, 1, 158, 482, 716): [0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35]
# (0, 4, 29, 309, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (1, 4, 204, 540, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (0, 1, 4, 172, 717): [0, 1, 3, 4, 7, 10, 13, 18, 23, 24, 25, 27, 28, 30, 31]
# (1, 29, 238, 563, 709): [1, 5, 6, 9, 10, 13, 14, 15, 17, 19, 21, 22, 24, 25, 27, 29, 31, 32, 34, 35]
# (0, 1, 29, 158, 716): [0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35]
# (0, 1, 29, 238, 563): [0, 1, 5, 10, 13, 14, 17, 29, 32, 34, 35]
# (0, 1, 4, 158, 717): [0, 1, 3, 4, 7, 10, 13, 18, 23, 24, 25, 27, 28, 30, 31]
# (0, 4, 29, 165, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (0, 1, 4, 158, 691): [0, 1, 4, 5, 6, 8, 11, 12, 15, 16, 18, 19, 20, 22, 23, 24, 25, 29, 32, 33]
# (0, 1, 4, 172, 692): [0, 1, 4, 6, 11, 12, 14, 15, 16, 17, 19, 20, 22, 27, 28, 29, 30, 31, 34, 35]
# (0, 4, 29, 309, 636): [0, 4, 5, 7, 8, 13, 17, 29, 32, 33, 35]
# (0, 1, 4, 204, 687): [0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 19, 20, 21, 22, 26, 29]
# (0, 4, 29, 309, 691): [0, 1, 4, 5, 6, 8, 11, 12, 15, 16, 18, 19, 20, 22, 23, 24, 25, 29, 32, 33]
# (0, 1, 29, 165, 490): [0, 1, 2, 5, 9, 21, 26, 27, 29, 30, 32]
# (1, 29, 238, 563, 705): [1, 2, 5, 10, 11, 12, 13, 14, 16, 17, 18, 20, 23, 26, 28, 29, 30, 32, 34, 35]
# (1, 4, 204, 543, 703): [1, 2, 3, 4, 6, 7, 10, 11, 13, 15, 20, 21, 24, 25, 30, 31, 32, 33, 34, 35]
# (0, 29, 165, 493, 699): [0, 2, 7, 8, 9, 10, 11, 15, 16, 17, 19, 21, 23, 25, 26, 28, 29, 31, 33, 35]
# (0, 29, 165, 490, 695): [0, 2, 3, 5, 6, 9, 12, 13, 17, 20, 21, 22, 23, 25, 26, 27, 29, 30, 32, 35]
# (0, 1, 29, 165, 716): [0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35]
# (0, 29, 165, 493, 696): [0, 2, 3, 6, 8, 9, 12, 13, 14, 18, 20, 21, 22, 24, 26, 28, 29, 31, 33, 34]
# (0, 1, 4, 204, 543): [0, 1, 3, 4, 7, 10, 13, 32, 33, 34, 35]
# (0, 1, 4, 158, 482): [0, 1, 4, 18, 21, 23, 24, 25, 26, 34, 35]
# (0, 1, 158, 473, 689): [0, 1, 2, 5, 9, 10, 11, 13, 14, 17, 18, 19, 20, 22, 23, 24, 25, 28, 31, 33]
# (0, 29, 165, 490, 698): [0, 2, 5, 7, 9, 10, 11, 14, 15, 16, 18, 19, 21, 24, 26, 27, 29, 30, 32, 34]
# (0, 4, 172, 506, 701): [0, 4, 5, 6, 7, 8, 11, 12, 13, 17, 18, 19, 21, 24, 26, 27, 28, 30, 31, 34]
# (0, 29, 165, 490, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (0, 29, 165, 493, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (4, 29, 309, 613, 712): [3, 4, 5, 8, 9, 10, 12, 14, 15, 19, 20, 21, 23, 24, 27, 28, 29, 32, 33, 34]
# (0, 4, 172, 487, 694): [0, 2, 3, 4, 6, 9, 10, 11, 12, 17, 19, 23, 25, 27, 28, 30, 31, 32, 33, 34]
# (1, 29, 238, 548, 708): [1, 3, 7, 8, 9, 11, 12, 14, 16, 17, 20, 21, 24, 25, 28, 29, 30, 33, 34, 35]
# (0, 4, 29, 172, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (1, 4, 204, 543, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (0, 1, 29, 165, 687): [0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 19, 20, 21, 22, 26, 29]
# (1, 4, 204, 543, 707): [1, 3, 4, 7, 9, 10, 12, 13, 16, 18, 19, 22, 23, 26, 27, 28, 32, 33, 34, 35]
# (0, 1, 158, 482, 690): [0, 1, 3, 5, 7, 11, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 33, 34, 35]
# (0, 1, 4, 172, 487): [0, 1, 2, 4, 9, 27, 28, 30, 31, 32, 33]
# (4, 29, 309, 613, 710): [2, 3, 4, 5, 6, 8, 10, 11, 14, 16, 18, 22, 25, 26, 29, 30, 31, 32, 33, 34]
# (4, 29, 309, 636, 711): [2, 4, 5, 7, 8, 12, 13, 15, 17, 19, 20, 23, 24, 26, 29, 30, 31, 32, 33, 35]
# (0, 1, 29, 238, 692): [0, 1, 4, 6, 11, 12, 14, 15, 16, 17, 19, 20, 22, 27, 28, 29, 30, 31, 34, 35]
# (4, 29, 309, 636, 713): [4, 5, 6, 7, 8, 9, 11, 13, 16, 17, 18, 21, 22, 25, 27, 28, 29, 32, 33, 35]
# (0, 4, 172, 487, 697): [0, 2, 4, 7, 9, 13, 14, 15, 16, 18, 20, 22, 24, 27, 28, 30, 31, 32, 33, 35]
# (0, 4, 172, 506, 722): [0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33]
# (0, 1, 4, 204, 540): [0, 1, 3, 4, 5, 7, 8, 10, 13, 14, 17]
# (1, 4, 29, 204, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (1, 4, 204, 540, 702): [1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 27, 28]
# (0, 1, 158, 482, 693): [0, 1, 6, 8, 10, 12, 13, 15, 16, 18, 21, 23, 24, 25, 26, 28, 31, 32, 34, 35]
# (0, 1, 29, 238, 548): [0, 1, 3, 7, 8, 14, 17, 29, 33, 34, 35]
# (1, 4, 29, 238, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (1, 29, 238, 548, 704): [1, 2, 3, 6, 7, 8, 14, 15, 17, 18, 19, 22, 23, 26, 27, 29, 31, 33, 34, 35]
# (4, 29, 309, 636, 732): [1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35]
# (1, 4, 204, 540, 706): [1, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 16, 17, 19, 22, 24, 25, 26, 30, 31]
# (0, 1, 158, 473, 688): [0, 1, 2, 3, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 23, 24, 25, 27, 30, 32]

# We group together all the keys with the same list of roots.
allRelevantRoots = list(set([frozenset(newrootsForConesaa2a3a4[u]) for u in newrootsForConesaa2a3a4.keys()]))

# len(allRelevantRoots)
# 43

allTuplesForRelevantRoots = dict()
for x in allRelevantRoots:
    allTuplesForRelevantRoots[tuple(sorted(list(x)))] = [u  for u in newrootsForConesaa2a3a4.keys() if newrootsForConesaa2a3a4[u] == sorted(list(x))]

# print [len(y) for y in allTuplesForRelevantRoots.values()]
# [1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 9, 1, 7, 1, 1]
# set([len(y) for y in allTuplesForRelevantRoots.values()])
# {1, 2, 3, 5, 7, 9}

len9 = [k for k in allTuplesForRelevantRoots.keys() if len(allTuplesForRelevantRoots[k])== 9]
# print len9
# [(1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35)]
# allTuplesForRelevantRoots[len9[0]]
# [(4, 29, 309, 613, 732),
#  (1, 29, 238, 563, 732),
#  (1, 4, 29, 309, 732),
#  (1, 29, 238, 548, 732),
#  (1, 4, 204, 540, 732),
#  (1, 4, 204, 543, 732),
#  (1, 4, 29, 204, 732),
#  (1, 4, 29, 238, 732),
#  (4, 29, 309, 636, 732)]

len7 = [k for k in allTuplesForRelevantRoots.keys() if len(allTuplesForRelevantRoots[k])== 7]
# print len7
# [(0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33)]
# allTuplesForRelevantRoots[len7[0]]
# [(0, 4, 172, 487, 722),
#  (0, 4, 29, 309, 722),
#  (0, 4, 29, 165, 722),
#  (0, 29, 165, 490, 722),
#  (0, 29, 165, 493, 722),
#  (0, 4, 29, 172, 722),
#  (0, 4, 172, 506, 722)]

len5 = [k for k in allTuplesForRelevantRoots.keys() if len(allTuplesForRelevantRoots[k])== 5]
# print len5
# [(0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35)]
# allTuplesForRelevantRoots[len5[0]]
# [(0, 1, 29, 238, 716),
#  (0, 1, 158, 473, 716),
#  (0, 1, 158, 482, 716),
#  (0, 1, 29, 158, 716),
#  (0, 1, 29, 165, 716)]

len3 = [k for k in allTuplesForRelevantRoots.keys() if len(allTuplesForRelevantRoots[k])== 3]
# print len3
# [(0, 1, 3, 4, 7, 10, 13, 18, 23, 24, 25, 27, 28, 30, 31)]
# allTuplesForRelevantRoots[len3[0]]
# [(0, 1, 4, 204, 717), (0, 1, 4, 172, 717), (0, 1, 4, 158, 717)]

len2 = [k for k in allTuplesForRelevantRoots.keys() if len(allTuplesForRelevantRoots[k])== 2]
# len2
# [(0, 1, 4, 5, 6, 8, 11, 12, 15, 16, 18, 19, 20, 22, 23, 24, 25, 29, 32, 33),
#  (0, 1, 4, 6, 11, 12, 14, 15, 16, 17, 19, 20, 22, 27, 28, 29, 30, 31, 34, 35),
#  (0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 19, 20, 21, 22, 26, 29)]
# allTuplesForRelevantRoots[len2[0]]
# [(0, 1, 4, 158, 691), (0, 4, 29, 309, 691)]
# allTuplesForRelevantRoots[len2[1]]
# [(0, 1, 4, 172, 692), (0, 1, 29, 238, 692)]
# allTuplesForRelevantRoots[len2[2]]
# [(0, 1, 4, 204, 687), (0, 1, 29, 165, 687)]

len1 = [k for k in allTuplesForRelevantRoots.keys() if len(allTuplesForRelevantRoots[k])== 1]
# len(len1)
# 36
# print len1
# [(0, 1, 2, 4, 9, 14, 17, 18, 23, 24, 25), (1, 5, 6, 9, 10, 13, 14, 15, 17, 19, 21, 22, 24, 25, 27, 29, 31, 32, 34, 35), (0, 1, 4, 18, 21, 23, 24, 25, 26, 34, 35), (0, 1, 2, 8, 9, 21, 26, 28, 29, 31, 33), (1, 3, 7, 8, 9, 11, 12, 14, 16, 17, 20, 21, 24, 25, 28, 29, 30, 33, 34, 35), (0, 1, 2, 3, 6, 7, 8, 9, 12, 14, 15, 16, 17, 18, 23, 24, 25, 27, 30, 32), (1, 2, 3, 4, 6, 7, 10, 11, 13, 15, 20, 21, 24, 25, 30, 31, 32, 33, 34, 35), (0, 4, 5, 7, 8, 13, 17, 29, 32, 33, 35), (4, 5, 6, 7, 8, 9, 11, 13, 16, 17, 18, 21, 22, 25, 27, 28, 29, 32, 33, 35), (0, 1, 3, 7, 8, 14, 17, 29, 33, 34, 35), (0, 1, 6, 8, 10, 12, 13, 15, 16, 18, 21, 23, 24, 25, 26, 28, 31, 32, 34, 35), (0, 1, 2, 4, 9, 27, 28, 30, 31, 32, 33), (0, 3, 4, 5, 8, 10, 14, 29, 32, 33, 34), (0, 1, 4, 5, 8, 21, 26, 27, 28, 30, 31), (0, 2, 3, 4, 6, 9, 10, 11, 12, 17, 19, 23, 25, 27, 28, 30, 31, 32, 33, 34), (1, 2, 3, 6, 7, 8, 14, 15, 17, 18, 19, 22, 23, 26, 27, 29, 31, 33, 34, 35), (1, 2, 5, 10, 11, 12, 13, 14, 16, 17, 18, 20, 23, 26, 28, 29, 30, 32, 34, 35), (0, 2, 4, 7, 9, 13, 14, 15, 16, 18, 20, 22, 24, 27, 28, 30, 31, 32, 33, 35), (3, 4, 5, 8, 9, 10, 12, 14, 15, 19, 20, 21, 23, 24, 27, 28, 29, 32, 33, 34), (0, 1, 3, 5, 7, 11, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 30, 33, 34, 35), (0, 3, 4, 5, 8, 10, 14, 15, 16, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 35), (0, 1, 3, 4, 5, 7, 8, 10, 13, 14, 17), (0, 1, 5, 10, 13, 14, 17, 29, 32, 34, 35), (1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 18, 20, 21, 23, 27, 28), (2, 3, 4, 5, 6, 8, 10, 11, 14, 16, 18, 22, 25, 26, 29, 30, 31, 32, 33, 34), (0, 4, 5, 6, 7, 8, 11, 12, 13, 17, 18, 19, 21, 24, 26, 27, 28, 30, 31, 34), (0, 2, 7, 8, 9, 10, 11, 15, 16, 17, 19, 21, 23, 25, 26, 28, 29, 31, 33, 35), (2, 4, 5, 7, 8, 12, 13, 15, 17, 19, 20, 23, 24, 26, 29, 30, 31, 32, 33, 35), (0, 1, 2, 5, 9, 10, 11, 13, 14, 17, 18, 19, 20, 22, 23, 24, 25, 28, 31, 33), (0, 2, 3, 5, 6, 9, 12, 13, 17, 20, 21, 22, 23, 25, 26, 27, 29, 30, 32, 35), (0, 1, 3, 4, 7, 10, 13, 32, 33, 34, 35), (1, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 16, 17, 19, 22, 24, 25, 26, 30, 31), (0, 2, 3, 6, 8, 9, 12, 13, 14, 18, 20, 21, 22, 24, 26, 28, 29, 31, 33, 34), (0, 1, 2, 5, 9, 21, 26, 27, 29, 30, 32), (0, 2, 5, 7, 9, 10, 11, 14, 15, 16, 18, 19, 21, 24, 26, 27, 29, 30, 32, 34), (1, 3, 4, 7, 9, 10, 12, 13, 16, 18, 19, 22, 23, 26, 27, 28, 32, 33, 34, 35)]
# {j: allTuplesForRelevantRoots[len1[j]] for j in range(0,len(len1))}
# {0: [(0, 1, 4, 158, 473)],
#  1: [(1, 29, 238, 563, 709)],
#  2: [(0, 1, 4, 158, 482)],
#  3: [(0, 1, 29, 165, 493)],
#  4: [(1, 29, 238, 548, 708)],
#  5: [(0, 1, 158, 473, 688)],
#  6: [(1, 4, 204, 543, 703)],
#  7: [(0, 4, 29, 309, 636)],
#  8: [(4, 29, 309, 636, 713)],
#  9: [(0, 1, 29, 238, 548)],
#  10: [(0, 1, 158, 482, 693)],
#  11: [(0, 1, 4, 172, 487)],
#  12: [(0, 4, 29, 309, 613)],
#  13: [(0, 1, 4, 172, 506)],
#  14: [(0, 4, 172, 487, 694)],
#  15: [(1, 29, 238, 548, 704)],
#  16: [(1, 29, 238, 563, 705)],
#  17: [(0, 4, 172, 487, 697)],
#  18: [(4, 29, 309, 613, 712)],
#  19: [(0, 1, 158, 482, 690)],
#  20: [(0, 4, 172, 506, 700)],
#  21: [(0, 1, 4, 204, 540)],
#  22: [(0, 1, 29, 238, 563)],
#  23: [(1, 4, 204, 540, 702)],
#  24: [(4, 29, 309, 613, 710)],
#  25: [(0, 4, 172, 506, 701)],
#  26: [(0, 29, 165, 493, 699)],
#  27: [(4, 29, 309, 636, 711)],
#  28: [(0, 1, 158, 473, 689)],
#  29: [(0, 29, 165, 490, 695)],
#  30: [(0, 1, 4, 204, 543)],
#  31: [(1, 4, 204, 540, 706)],
#  32: [(0, 29, 165, 493, 696)],
#  33: [(0, 1, 29, 165, 490)],
#  34: [(0, 29, 165, 490, 698)],
#  35: [(1, 4, 204, 543, 707)]}



###############################
# Relevant Roots For Summands #
###############################

rootsForConesaa2a3a4SummandY34 = dict()
rootsForConesaa2a3a4SummandY8 = dict()
for u in raysFibersInaa2a3a4_lower_dim.keys():
    rootsForConesaa2a3a4SummandY34[u] = [x for x in rootsForConesaa2a3a4[u] if x in [24, 10, 17, 27, 26, 33]]
    rootsForConesaa2a3a4SummandY8[u] = [x for x in rootsForConesaa2a3a4[u] if x in [9, 5, 7, 23, 31, 34]]

# rootsForConesaa2a3a4SummandY34
# {(0, 1, 4, 158, 473): [17, 24],
#  (0, 1, 4, 158, 482): [24, 26],
#  (0, 1, 4, 158, 691): [24, 33],
#  (0, 1, 4, 158, 717): [10, 24, 27],
#  (0, 1, 4, 172, 487): [27, 33],
#  (0, 1, 4, 172, 506): [26, 27],
#  (0, 1, 4, 172, 692): [17, 27],
#  (0, 1, 4, 172, 717): [10, 24, 27],
#  (0, 1, 4, 204, 540): [10, 17],
#  (0, 1, 4, 204, 543): [10, 33],
#  (0, 1, 4, 204, 687): [10, 26],
#  (0, 1, 4, 204, 717): [10, 24, 27],
#  (0, 1, 29, 158, 716): [17, 24, 26],
#  (0, 1, 29, 165, 490): [26, 27],
#  (0, 1, 29, 165, 493): [26, 33],
#  (0, 1, 29, 165, 687): [10, 26],
#  (0, 1, 29, 165, 716): [17, 24, 26],
#  (0, 1, 29, 238, 548): [17, 33],
#  (0, 1, 29, 238, 563): [10, 17],
#  (0, 1, 29, 238, 692): [17, 27],
#  (0, 1, 29, 238, 716): [17, 24, 26],
#  (0, 1, 158, 473, 688): [17, 24, 27],
#  (0, 1, 158, 473, 689): [10, 17, 24, 33],
#  (0, 1, 158, 473, 716): [17, 24, 26],
#  (0, 1, 158, 482, 690): [24, 26, 27, 33],
#  (0, 1, 158, 482, 693): [10, 24, 26],
#  (0, 1, 158, 482, 716): [17, 24, 26],
#  (0, 4, 29, 165, 722): [26, 27, 33],
#  (0, 4, 29, 172, 722): [26, 27, 33],
#  (0, 4, 29, 309, 613): [10, 33],
#  (0, 4, 29, 309, 636): [17, 33],
#  (0, 4, 29, 309, 691): [24, 33],
#  (0, 4, 29, 309, 722): [26, 27, 33],
#  (0, 4, 172, 487, 694): [10, 17, 27, 33],
#  (0, 4, 172, 487, 697): [24, 27, 33],
#  (0, 4, 172, 487, 722): [26, 27, 33],
#  (0, 4, 172, 506, 700): [10, 26, 27],
#  (0, 4, 172, 506, 701): [17, 24, 26, 27],
#  (0, 4, 172, 506, 722): [26, 27, 33],
#  (0, 29, 165, 490, 695): [17, 26, 27],
#  (0, 29, 165, 490, 698): [10, 24, 26, 27],
#  (0, 29, 165, 490, 722): [26, 27, 33],
#  (0, 29, 165, 493, 696): [24, 26, 33],
#  (0, 29, 165, 493, 699): [10, 17, 26, 33],
#  (0, 29, 165, 493, 722): [26, 27, 33],
#  (1, 4, 29, 204, 732): [10, 17, 33],
#  (1, 4, 29, 238, 732): [10, 17, 33],
#  (1, 4, 29, 309, 732): [10, 17, 33],
#  (1, 4, 204, 540, 702): [10, 17, 27],
#  (1, 4, 204, 540, 706): [10, 17, 24, 26],
#  (1, 4, 204, 540, 732): [10, 17, 33],
#  (1, 4, 204, 543, 703): [10, 24, 33],
#  (1, 4, 204, 543, 707): [10, 26, 27, 33],
#  (1, 4, 204, 543, 732): [10, 17, 33],
#  (1, 29, 238, 548, 704): [17, 26, 27, 33],
#  (1, 29, 238, 548, 708): [17, 24, 33],
#  (1, 29, 238, 548, 732): [10, 17, 33],
#  (1, 29, 238, 563, 705): [10, 17, 26],
#  (1, 29, 238, 563, 709): [10, 17, 24, 27],
#  (1, 29, 238, 563, 732): [10, 17, 33],
#  (4, 29, 309, 613, 710): [10, 26, 33],
#  (4, 29, 309, 613, 712): [10, 24, 27, 33],
#  (4, 29, 309, 613, 732): [10, 17, 33],
#  (4, 29, 309, 636, 711): [17, 24, 26, 33],
#  (4, 29, 309, 636, 713): [17, 27, 33],
#  (4, 29, 309, 636, 732): [10, 17, 33]}

# rootsForConesaa2a3a4SummandY8
# {(0, 1, 4, 158, 473): [9, 23],
#  (0, 1, 4, 158, 482): [23, 34],
#  (0, 1, 4, 158, 691): [5, 23],
#  (0, 1, 4, 158, 717): [7, 23, 31],
#  (0, 1, 4, 172, 487): [9, 31],
#  (0, 1, 4, 172, 506): [5, 31],
#  (0, 1, 4, 172, 692): [31, 34],
#  (0, 1, 4, 172, 717): [7, 23, 31],
#  (0, 1, 4, 204, 540): [5, 7],
#  (0, 1, 4, 204, 543): [7, 34],
#  (0, 1, 4, 204, 687): [7, 9],
#  (0, 1, 4, 204, 717): [7, 23, 31],
#  (0, 1, 29, 158, 716): [9, 23, 34],
#  (0, 1, 29, 165, 490): [5, 9],
#  (0, 1, 29, 165, 493): [9, 31],
#  (0, 1, 29, 165, 687): [7, 9],
#  (0, 1, 29, 165, 716): [9, 23, 34],
#  (0, 1, 29, 238, 548): [7, 34],
#  (0, 1, 29, 238, 563): [5, 34],
#  (0, 1, 29, 238, 692): [31, 34],
#  (0, 1, 29, 238, 716): [9, 23, 34],
#  (0, 1, 158, 473, 688): [7, 9, 23],
#  (0, 1, 158, 473, 689): [5, 9, 23, 31],
#  (0, 1, 158, 473, 716): [9, 23, 34],
#  (0, 1, 158, 482, 690): [5, 7, 23, 34],
#  (0, 1, 158, 482, 693): [23, 31, 34],
#  (0, 1, 158, 482, 716): [9, 23, 34],
#  (0, 4, 29, 165, 722): [5, 9, 31],
#  (0, 4, 29, 172, 722): [5, 9, 31],
#  (0, 4, 29, 309, 613): [5, 34],
#  (0, 4, 29, 309, 636): [5, 7],
#  (0, 4, 29, 309, 691): [5, 23],
#  (0, 4, 29, 309, 722): [5, 9, 31],
#  (0, 4, 172, 487, 694): [9, 23, 31, 34],
#  (0, 4, 172, 487, 697): [7, 9, 31],
#  (0, 4, 172, 487, 722): [5, 9, 31],
#  (0, 4, 172, 506, 700): [5, 23, 31],
#  (0, 4, 172, 506, 701): [5, 7, 31, 34],
#  (0, 4, 172, 506, 722): [5, 9, 31],
#  (0, 29, 165, 490, 695): [5, 9, 23],
#  (0, 29, 165, 490, 698): [5, 7, 9, 34],
#  (0, 29, 165, 490, 722): [5, 9, 31],
#  (0, 29, 165, 493, 696): [9, 31, 34],
#  (0, 29, 165, 493, 699): [7, 9, 23, 31],
#  (0, 29, 165, 493, 722): [5, 9, 31],
#  (1, 4, 29, 204, 732): [5, 7, 34],
#  (1, 4, 29, 238, 732): [5, 7, 34],
#  (1, 4, 29, 309, 732): [5, 7, 34],
#  (1, 4, 204, 540, 702): [5, 7, 23],
#  (1, 4, 204, 540, 706): [5, 7, 9, 31],
#  (1, 4, 204, 540, 732): [5, 7, 34],
#  (1, 4, 204, 543, 703): [7, 31, 34],
#  (1, 4, 204, 543, 707): [7, 9, 23, 34],
#  (1, 4, 204, 543, 732): [5, 7, 34],
#  (1, 29, 238, 548, 704): [7, 23, 31, 34],
#  (1, 29, 238, 548, 708): [7, 9, 34],
#  (1, 29, 238, 548, 732): [5, 7, 34],
#  (1, 29, 238, 563, 705): [5, 23, 34],
#  (1, 29, 238, 563, 709): [5, 9, 31, 34],
#  (1, 29, 238, 563, 732): [5, 7, 34],
#  (4, 29, 309, 613, 710): [5, 31, 34],
#  (4, 29, 309, 613, 712): [5, 9, 23, 34],
#  (4, 29, 309, 613, 732): [5, 7, 34],
#  (4, 29, 309, 636, 711): [5, 7, 23, 31],
#  (4, 29, 309, 636, 713): [5, 7, 9],
#  (4, 29, 309, 636, 732): [5, 7, 34]}


rootValuesaa2a3a4 = dict()
for u in raysFibersInaa2a3a4_lower_dim.keys():
    rootValuesaa2a3a4[u] = dict()
    l = len(raysFibersInaa2a3a4_lower_dim[u])
    rootValuesaa2a3a4[u] = {r: tuple([raysFibersInaa2a3a4_lower_dim[u][k][r] for k in range(0,l)]) for r in rootsForConesaa2a3a4[u]}
    
# We record the output:
# rootValuesaa2a3a4
# {(0, 1, 4, 158, 473): {0: (1, 2, 1, 3, 1),
#   1: (0, 2, 1, 3, 1),
#   2: (0, 0, 0, 1, 0),
#   4: (0, 0, 0, 0, 1),
#   9: (0, 0, 0, 1, 0),
#   14: (0, 0, 0, 1, 0),
#   17: (0, 0, 0, 1, 0),
#   18: (0, 1, 0, 1, 0),
#   23: (0, 1, 0, 1, 0),
#   24: (0, 1, 0, 1, 0),
#   25: (0, 1, 0, 1, 0)},
#  (0, 1, 4, 158, 482): {0: (1, 2, 1, 3, 1),
#   1: (0, 2, 1, 3, 1),
#   4: (0, 0, 0, 0, 1),
#   18: (0, 1, 0, 1, 0),
#   21: (0, 0, 0, 1, 0),
#   23: (0, 1, 0, 1, 0),
#   24: (0, 1, 0, 1, 0),
#   25: (0, 1, 0, 1, 0),
#   26: (0, 0, 0, 1, 0),
#   34: (0, 0, 0, 1, 0),
#   35: (0, 0, 0, 1, 0)},
#  (0, 1, 4, 158, 691): {0: (1, 1, 2, 1, 1),
#   1: (1, 0, 2, 1, 1),
#   4: (1, 0, 0, 0, 1),
#   5: (1, 0, 0, 0, 0),
#   6: (1, 0, 0, 0, 0),
#   8: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   18: (1, 0, 1, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 1, 0, 0),
#   24: (1, 0, 1, 0, 0),
#   25: (1, 0, 1, 0, 0),
#   29: (1, 0, 0, 0, 0),
#   32: (1, 0, 0, 0, 0),
#   33: (1, 0, 0, 0, 0)},
#  (0, 1, 4, 158, 717): {0: (1, 1, 2, 4, 1),
#   1: (0, 1, 2, 4, 1),
#   3: (0, 0, 0, 1, 0),
#   4: (0, 0, 0, 4, 1),
#   7: (0, 0, 0, 1, 0),
#   10: (0, 0, 0, 1, 0),
#   13: (0, 0, 0, 1, 0),
#   18: (0, 0, 1, 1, 0),
#   23: (0, 0, 1, 1, 0),
#   24: (0, 0, 1, 1, 0),
#   25: (0, 0, 1, 1, 0),
#   27: (0, 0, 0, 1, 0),
#   28: (0, 0, 0, 1, 0),
#   30: (0, 0, 0, 1, 0),
#   31: (0, 0, 0, 1, 0)},
#  (0, 1, 4, 172, 487): {0: (2, 1, 1, 1, 3, 3),
#   1: (0, 0, 1, 1, 0, 0),
#   2: (0, 0, 0, 0, 1, 1),
#   4: (2, 0, 0, 1, 3, 1),
#   9: (0, 0, 0, 0, 1, 1),
#   27: (1, 0, 0, 0, 1, 1),
#   28: (1, 0, 0, 0, 1, 1),
#   30: (1, 0, 0, 0, 1, 1),
#   31: (1, 0, 0, 0, 1, 1),
#   32: (0, 0, 0, 0, 1, 1),
#   33: (0, 0, 0, 0, 1, 1)},
#  (0, 1, 4, 172, 506): {0: (2, 1, 1, 1, 3, 3),
#   1: (0, 0, 1, 1, 0, 0),
#   4: (2, 0, 0, 1, 3, 1),
#   5: (0, 0, 0, 0, 1, 1),
#   8: (0, 0, 0, 0, 1, 1),
#   21: (0, 0, 0, 0, 1, 1),
#   26: (0, 0, 0, 0, 1, 1),
#   27: (1, 0, 0, 0, 1, 1),
#   28: (1, 0, 0, 0, 1, 1),
#   30: (1, 0, 0, 0, 1, 1),
#   31: (1, 0, 0, 0, 1, 1)},
#  (0, 1, 4, 172, 692): {0: (1, 1, 2, 1, 1),
#   1: (1, 0, 0, 1, 1),
#   4: (1, 0, 2, 1, 0),
#   6: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   14: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   27: (1, 0, 1, 0, 0),
#   28: (1, 0, 1, 0, 0),
#   29: (1, 0, 0, 0, 0),
#   30: (1, 0, 1, 0, 0),
#   31: (1, 0, 1, 0, 0),
#   34: (1, 0, 0, 0, 0),
#   35: (1, 0, 0, 0, 0)},
#  (0, 1, 4, 172, 717): {0: (1, 2, 1, 4, 1),
#   1: (0, 0, 1, 4, 1),
#   3: (0, 0, 0, 1, 0),
#   4: (0, 2, 1, 4, 0),
#   7: (0, 0, 0, 1, 0),
#   10: (0, 0, 0, 1, 0),
#   13: (0, 0, 0, 1, 0),
#   18: (0, 0, 0, 1, 0),
#   23: (0, 0, 0, 1, 0),
#   24: (0, 0, 0, 1, 0),
#   25: (0, 0, 0, 1, 0),
#   27: (0, 1, 0, 1, 0),
#   28: (0, 1, 0, 1, 0),
#   30: (0, 1, 0, 1, 0),
#   31: (0, 1, 0, 1, 0)},
#  (0, 1, 4, 204, 540): {0: (0, 1, 1, 1, 0, 0, 0),
#   1: (2, 0, 1, 1, 3, 1, 3),
#   3: (1, 0, 0, 0, 1, 1, 1),
#   4: (2, 0, 0, 1, 3, 1, 1),
#   5: (0, 0, 0, 0, 1, 1, 1),
#   7: (1, 0, 0, 0, 1, 1, 1),
#   8: (0, 0, 0, 0, 1, 1, 1),
#   10: (1, 0, 0, 0, 1, 1, 1),
#   13: (1, 0, 0, 0, 1, 1, 1),
#   14: (0, 0, 0, 0, 1, 1, 1),
#   17: (0, 0, 0, 0, 1, 1, 1)},
#  (0, 1, 4, 204, 543): {0: (0, 1, 1, 1, 0, 0, 0),
#   1: (2, 0, 1, 1, 3, 1, 3),
#   3: (1, 0, 0, 0, 1, 1, 1),
#   4: (2, 0, 0, 1, 3, 1, 1),
#   7: (1, 0, 0, 0, 1, 1, 1),
#   10: (1, 0, 0, 0, 1, 1, 1),
#   13: (1, 0, 0, 0, 1, 1, 1),
#   32: (0, 0, 0, 0, 1, 1, 1),
#   33: (0, 0, 0, 0, 1, 1, 1),
#   34: (0, 0, 0, 0, 1, 1, 1),
#   35: (0, 0, 0, 0, 1, 1, 1)},
#  (0, 1, 4, 204, 687): {0: (1, 0, 1, 1, 1),
#   1: (1, 2, 1, 1, 0),
#   2: (1, 0, 0, 0, 0),
#   3: (1, 1, 0, 0, 0),
#   4: (1, 2, 0, 1, 0),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 1, 0, 0, 0),
#   9: (1, 0, 0, 0, 0),
#   10: (1, 1, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 1, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   26: (1, 0, 0, 0, 0),
#   29: (1, 0, 0, 0, 0)},
#  (0, 1, 4, 204, 717): {0: (0, 1, 1, 4, 1),
#   1: (2, 1, 1, 4, 0),
#   3: (1, 0, 0, 1, 0),
#   4: (2, 0, 1, 4, 0),
#   7: (1, 0, 0, 1, 0),
#   10: (1, 0, 0, 1, 0),
#   13: (1, 0, 0, 1, 0),
#   18: (0, 0, 0, 1, 0),
#   23: (0, 0, 0, 1, 0),
#   24: (0, 0, 0, 1, 0),
#   25: (0, 0, 0, 1, 0),
#   27: (0, 0, 0, 1, 0),
#   28: (0, 0, 0, 1, 0),
#   30: (0, 0, 0, 1, 0),
#   31: (0, 0, 0, 1, 0)},
#  (0, 1, 29, 158, 716): {0: (1, 1, 4, 4, 2),
#   1: (0, 1, 4, 4, 2),
#   2: (0, 0, 1, 1, 0),
#   9: (0, 0, 1, 1, 0),
#   14: (0, 0, 1, 1, 0),
#   17: (0, 0, 1, 1, 0),
#   18: (0, 0, 1, 1, 1),
#   21: (0, 0, 1, 1, 0),
#   23: (0, 0, 1, 1, 1),
#   24: (0, 0, 1, 1, 1),
#   25: (0, 0, 1, 1, 1),
#   26: (0, 0, 1, 1, 0),
#   29: (0, 0, 1, 4, 0),
#   34: (0, 0, 1, 1, 0),
#   35: (0, 0, 1, 1, 0)},
#  (0, 1, 29, 165, 490): {0: (1, 2, 2, 3, 1),
#   1: (0, 0, 0, 0, 1),
#   2: (0, 1, 1, 1, 0),
#   5: (0, 0, 0, 1, 0),
#   9: (0, 1, 1, 1, 0),
#   21: (0, 1, 1, 1, 0),
#   26: (0, 1, 1, 1, 0),
#   27: (0, 0, 0, 1, 0),
#   29: (0, 2, 1, 1, 0),
#   30: (0, 0, 0, 1, 0),
#   32: (0, 0, 0, 1, 0)},
#  (0, 1, 29, 165, 493): {0: (1, 2, 2, 3, 1),
#   1: (0, 0, 0, 0, 1),
#   2: (0, 1, 1, 1, 0),
#   8: (0, 0, 0, 1, 0),
#   9: (0, 1, 1, 1, 0),
#   21: (0, 1, 1, 1, 0),
#   26: (0, 1, 1, 1, 0),
#   28: (0, 0, 0, 1, 0),
#   29: (0, 2, 1, 1, 0),
#   31: (0, 0, 0, 1, 0),
#   33: (0, 0, 0, 1, 0)},
#  (0, 1, 29, 165, 687): {0: (1, 1, 1, 2, 2),
#   1: (1, 0, 1, 0, 0),
#   2: (1, 0, 0, 1, 1),
#   3: (1, 0, 0, 0, 0),
#   4: (1, 0, 0, 0, 0),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 0, 0, 0, 0),
#   9: (1, 0, 0, 1, 1),
#   10: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 0, 1, 1),
#   22: (1, 0, 0, 0, 0),
#   26: (1, 0, 0, 1, 1),
#   29: (1, 0, 0, 2, 1)},
#  (0, 1, 29, 165, 716): {0: (1, 2, 2, 1, 4, 4),
#   1: (0, 0, 0, 1, 4, 4),
#   2: (0, 1, 1, 0, 1, 1),
#   9: (0, 1, 1, 0, 1, 1),
#   14: (0, 0, 0, 0, 1, 1),
#   17: (0, 0, 0, 0, 1, 1),
#   18: (0, 0, 0, 0, 1, 1),
#   21: (0, 1, 1, 0, 1, 1),
#   23: (0, 0, 0, 0, 1, 1),
#   24: (0, 0, 0, 0, 1, 1),
#   25: (0, 0, 0, 0, 1, 1),
#   26: (0, 1, 1, 0, 1, 1),
#   29: (0, 2, 1, 0, 4, 1),
#   34: (0, 0, 0, 0, 1, 1),
#   35: (0, 0, 0, 0, 1, 1)},
#  (0, 1, 29, 238, 548): {0: (0, 0, 1, 1, 0, 0),
#   1: (2, 2, 1, 0, 1, 3),
#   3: (0, 0, 0, 0, 1, 1),
#   7: (0, 0, 0, 0, 1, 1),
#   8: (0, 0, 0, 0, 1, 1),
#   14: (1, 1, 0, 0, 1, 1),
#   17: (1, 1, 0, 0, 1, 1),
#   29: (1, 2, 0, 0, 1, 1),
#   33: (0, 0, 0, 0, 1, 1),
#   34: (1, 1, 0, 0, 1, 1),
#   35: (1, 1, 0, 0, 1, 1)},
#  (0, 1, 29, 238, 563): {0: (0, 0, 1, 1, 0, 0),
#   1: (2, 2, 1, 0, 1, 3),
#   5: (0, 0, 0, 0, 1, 1),
#   10: (0, 0, 0, 0, 1, 1),
#   13: (0, 0, 0, 0, 1, 1),
#   14: (1, 1, 0, 0, 1, 1),
#   17: (1, 1, 0, 0, 1, 1),
#   29: (1, 2, 0, 0, 1, 1),
#   32: (0, 0, 0, 0, 1, 1),
#   34: (1, 1, 0, 0, 1, 1),
#   35: (1, 1, 0, 0, 1, 1)},
#  (0, 1, 29, 238, 692): {0: (1, 0, 0, 1, 1),
#   1: (1, 2, 2, 0, 1),
#   4: (1, 0, 0, 0, 0),
#   6: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   14: (1, 1, 1, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   27: (1, 0, 0, 0, 0),
#   28: (1, 0, 0, 0, 0),
#   29: (1, 2, 1, 0, 0),
#   30: (1, 0, 0, 0, 0),
#   31: (1, 0, 0, 0, 0),
#   34: (1, 1, 1, 0, 0),
#   35: (1, 1, 1, 0, 0)},
#  (0, 1, 29, 238, 716): {0: (0, 0, 4, 1, 1, 4),
#   1: (2, 2, 4, 0, 1, 4),
#   2: (0, 0, 1, 0, 0, 1),
#   9: (0, 0, 1, 0, 0, 1),
#   14: (1, 1, 1, 0, 0, 1),
#   17: (1, 1, 1, 0, 0, 1),
#   18: (0, 0, 1, 0, 0, 1),
#   21: (0, 0, 1, 0, 0, 1),
#   23: (0, 0, 1, 0, 0, 1),
#   24: (0, 0, 1, 0, 0, 1),
#   25: (0, 0, 1, 0, 0, 1),
#   26: (0, 0, 1, 0, 0, 1),
#   29: (2, 1, 4, 0, 0, 1),
#   34: (1, 1, 1, 0, 0, 1),
#   35: (1, 1, 1, 0, 0, 1)},
#  (0, 1, 158, 473, 688): {0: (1, 1, 3, 1, 2),
#   1: (1, 0, 3, 1, 2),
#   2: (1, 0, 1, 0, 0),
#   3: (1, 0, 0, 0, 0),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 0, 0, 0, 0),
#   8: (1, 0, 0, 0, 0),
#   9: (1, 0, 1, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   14: (1, 0, 1, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 0, 1, 0, 0),
#   18: (1, 0, 1, 0, 1),
#   23: (1, 0, 1, 0, 1),
#   24: (1, 0, 1, 0, 1),
#   25: (1, 0, 1, 0, 1),
#   27: (1, 0, 0, 0, 0),
#   30: (1, 0, 0, 0, 0),
#   32: (1, 0, 0, 0, 0)},
#  (0, 1, 158, 473, 689): {0: (1, 1, 3, 1, 2),
#   1: (1, 0, 3, 1, 2),
#   2: (1, 0, 1, 0, 0),
#   5: (1, 0, 0, 0, 0),
#   9: (1, 0, 1, 0, 0),
#   10: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   14: (1, 0, 1, 0, 0),
#   17: (1, 0, 1, 0, 0),
#   18: (1, 0, 1, 0, 1),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 1, 0, 1),
#   24: (1, 0, 1, 0, 1),
#   25: (1, 0, 1, 0, 1),
#   28: (1, 0, 0, 0, 0),
#   31: (1, 0, 0, 0, 0),
#   33: (1, 0, 0, 0, 0)},
#  (0, 1, 158, 473, 716): {0: (1, 2, 1, 4, 3),
#   1: (0, 2, 1, 4, 3),
#   2: (0, 0, 0, 1, 1),
#   9: (0, 0, 0, 1, 1),
#   14: (0, 0, 0, 1, 1),
#   17: (0, 0, 0, 1, 1),
#   18: (0, 1, 0, 1, 1),
#   21: (0, 0, 0, 1, 0),
#   23: (0, 1, 0, 1, 1),
#   24: (0, 1, 0, 1, 1),
#   25: (0, 1, 0, 1, 1),
#   26: (0, 0, 0, 1, 0),
#   29: (0, 0, 0, 1, 0),
#   34: (0, 0, 0, 1, 0),
#   35: (0, 0, 0, 1, 0)},
#  (0, 1, 158, 482, 690): {0: (1, 1, 3, 1, 2),
#   1: (1, 0, 3, 1, 2),
#   3: (1, 0, 0, 0, 0),
#   5: (1, 0, 0, 0, 0),
#   7: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   18: (1, 0, 1, 0, 1),
#   19: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 1, 0, 1),
#   24: (1, 0, 1, 0, 1),
#   25: (1, 0, 1, 0, 1),
#   26: (1, 0, 1, 0, 0),
#   27: (1, 0, 0, 0, 0),
#   30: (1, 0, 0, 0, 0),
#   33: (1, 0, 0, 0, 0),
#   34: (1, 0, 1, 0, 0),
#   35: (1, 0, 1, 0, 0)},
#  (0, 1, 158, 482, 693): {0: (1, 1, 3, 1, 2),
#   1: (1, 0, 3, 1, 2),
#   6: (1, 0, 0, 0, 0),
#   8: (1, 0, 0, 0, 0),
#   10: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   18: (1, 0, 1, 0, 1),
#   21: (1, 0, 1, 0, 0),
#   23: (1, 0, 1, 0, 1),
#   24: (1, 0, 1, 0, 1),
#   25: (1, 0, 1, 0, 1),
#   26: (1, 0, 1, 0, 0),
#   28: (1, 0, 0, 0, 0),
#   31: (1, 0, 0, 0, 0),
#   32: (1, 0, 0, 0, 0),
#   34: (1, 0, 1, 0, 0),
#   35: (1, 0, 1, 0, 0)},
#  (0, 1, 158, 482, 716): {0: (1, 2, 1, 4, 3),
#   1: (0, 2, 1, 4, 3),
#   2: (0, 0, 0, 1, 0),
#   9: (0, 0, 0, 1, 0),
#   14: (0, 0, 0, 1, 0),
#   17: (0, 0, 0, 1, 0),
#   18: (0, 1, 0, 1, 1),
#   21: (0, 0, 0, 1, 1),
#   23: (0, 1, 0, 1, 1),
#   24: (0, 1, 0, 1, 1),
#   25: (0, 1, 0, 1, 1),
#   26: (0, 0, 0, 1, 1),
#   29: (0, 0, 0, 1, 0),
#   34: (0, 0, 0, 1, 1),
#   35: (0, 0, 0, 1, 1)},
#  (0, 4, 29, 165, 722): {0: (1, 2, 4, 2, 4, 4),
#   2: (0, 1, 1, 1, 1, 1),
#   4: (0, 0, 1, 0, 4, 4),
#   5: (0, 0, 1, 0, 1, 1),
#   8: (0, 0, 1, 0, 1, 1),
#   9: (0, 1, 1, 1, 1, 1),
#   21: (0, 1, 1, 1, 1, 1),
#   26: (0, 1, 1, 1, 1, 1),
#   27: (0, 0, 1, 0, 1, 1),
#   28: (0, 0, 1, 0, 1, 1),
#   29: (0, 1, 1, 2, 1, 4),
#   30: (0, 0, 1, 0, 1, 1),
#   31: (0, 0, 1, 0, 1, 1),
#   32: (0, 0, 1, 0, 1, 1),
#   33: (0, 0, 1, 0, 1, 1)},
#  (0, 4, 29, 172, 722): {0: (1, 4, 4, 2, 4),
#   2: (0, 1, 1, 0, 1),
#   4: (0, 1, 4, 2, 4),
#   5: (0, 1, 1, 0, 1),
#   8: (0, 1, 1, 0, 1),
#   9: (0, 1, 1, 0, 1),
#   21: (0, 1, 1, 0, 1),
#   26: (0, 1, 1, 0, 1),
#   27: (0, 1, 1, 1, 1),
#   28: (0, 1, 1, 1, 1),
#   29: (0, 1, 1, 0, 4),
#   30: (0, 1, 1, 1, 1),
#   31: (0, 1, 1, 1, 1),
#   32: (0, 1, 1, 0, 1),
#   33: (0, 1, 1, 0, 1)},
#  (0, 4, 29, 309, 613): {0: (0, 0, 0, 0, 1),
#   3: (0, 0, 0, 1, 0),
#   4: (1, 2, 2, 1, 0),
#   5: (1, 1, 1, 1, 0),
#   8: (1, 1, 1, 1, 0),
#   10: (0, 0, 0, 1, 0),
#   14: (0, 0, 0, 1, 0),
#   29: (1, 2, 1, 1, 0),
#   32: (1, 1, 1, 1, 0),
#   33: (1, 1, 1, 1, 0),
#   34: (0, 0, 0, 1, 0)},
#  (0, 4, 29, 309, 636): {0: (0, 0, 0, 0, 1),
#   4: (1, 2, 2, 1, 0),
#   5: (1, 1, 1, 1, 0),
#   7: (0, 0, 0, 1, 0),
#   8: (1, 1, 1, 1, 0),
#   13: (0, 0, 0, 1, 0),
#   17: (0, 0, 0, 1, 0),
#   29: (1, 2, 1, 1, 0),
#   32: (1, 1, 1, 1, 0),
#   33: (1, 1, 1, 1, 0),
#   35: (0, 0, 0, 1, 0)},
#  (0, 4, 29, 309, 691): {0: (1, 1, 0, 0, 0),
#   1: (0, 1, 0, 0, 0),
#   4: (0, 1, 1, 2, 2),
#   5: (0, 1, 1, 1, 1),
#   6: (0, 1, 0, 0, 0),
#   8: (0, 1, 1, 1, 1),
#   11: (0, 1, 0, 0, 0),
#   12: (0, 1, 0, 0, 0),
#   15: (0, 1, 0, 0, 0),
#   16: (0, 1, 0, 0, 0),
#   18: (0, 1, 0, 0, 0),
#   19: (0, 1, 0, 0, 0),
#   20: (0, 1, 0, 0, 0),
#   22: (0, 1, 0, 0, 0),
#   23: (0, 1, 0, 0, 0),
#   24: (0, 1, 0, 0, 0),
#   25: (0, 1, 0, 0, 0),
#   29: (0, 1, 1, 2, 1),
#   32: (0, 1, 1, 1, 1),
#   33: (0, 1, 1, 1, 1)},
#  (0, 4, 29, 309, 722): {0: (0, 0, 0, 4, 4, 4, 1),
#   2: (0, 0, 0, 1, 1, 1, 0),
#   4: (1, 2, 2, 4, 1, 4, 0),
#   5: (1, 1, 1, 1, 1, 1, 0),
#   8: (1, 1, 1, 1, 1, 1, 0),
#   9: (0, 0, 0, 1, 1, 1, 0),
#   21: (0, 0, 0, 1, 1, 1, 0),
#   26: (0, 0, 0, 1, 1, 1, 0),
#   27: (0, 0, 0, 1, 1, 1, 0),
#   28: (0, 0, 0, 1, 1, 1, 0),
#   29: (1, 1, 2, 4, 1, 1, 0),
#   30: (0, 0, 0, 1, 1, 1, 0),
#   31: (0, 0, 0, 1, 1, 1, 0),
#   32: (1, 1, 1, 1, 1, 1, 0),
#   33: (1, 1, 1, 1, 1, 1, 0)},
#  (0, 4, 172, 487, 694): {0: (1, 1, 3, 3, 2),
#   2: (1, 0, 1, 1, 0),
#   3: (1, 0, 0, 0, 0),
#   4: (1, 0, 1, 3, 2),
#   6: (1, 0, 0, 0, 0),
#   9: (1, 0, 1, 1, 0),
#   10: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   17: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   27: (1, 0, 1, 1, 1),
#   28: (1, 0, 1, 1, 1),
#   30: (1, 0, 1, 1, 1),
#   31: (1, 0, 1, 1, 1),
#   32: (1, 0, 1, 1, 0),
#   33: (1, 0, 1, 1, 0),
#   34: (1, 0, 0, 0, 0)},
#  (0, 4, 172, 487, 697): {0: (1, 1, 3, 3, 2),
#   2: (1, 0, 1, 1, 0),
#   4: (1, 0, 1, 3, 2),
#   7: (1, 0, 0, 0, 0),
#   9: (1, 0, 1, 1, 0),
#   13: (1, 0, 0, 0, 0),
#   14: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   18: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   24: (1, 0, 0, 0, 0),
#   27: (1, 0, 1, 1, 1),
#   28: (1, 0, 1, 1, 1),
#   30: (1, 0, 1, 1, 1),
#   31: (1, 0, 1, 1, 1),
#   32: (1, 0, 1, 1, 0),
#   33: (1, 0, 1, 1, 0),
#   35: (1, 0, 0, 0, 0)},
#  (0, 4, 172, 487, 722): {0: (1, 3, 4, 3, 2, 4),
#   2: (0, 1, 1, 1, 0, 1),
#   4: (0, 1, 1, 3, 2, 4),
#   5: (0, 0, 1, 0, 0, 1),
#   8: (0, 0, 1, 0, 0, 1),
#   9: (0, 1, 1, 1, 0, 1),
#   21: (0, 0, 1, 0, 0, 1),
#   26: (0, 0, 1, 0, 0, 1),
#   27: (0, 1, 1, 1, 1, 1),
#   28: (0, 1, 1, 1, 1, 1),
#   29: (0, 0, 1, 0, 0, 1),
#   30: (0, 1, 1, 1, 1, 1),
#   31: (0, 1, 1, 1, 1, 1),
#   32: (0, 1, 1, 1, 0, 1),
#   33: (0, 1, 1, 1, 0, 1)},
#  (0, 4, 172, 506, 700): {0: (1, 1, 3, 3, 2),
#   3: (1, 0, 0, 0, 0),
#   4: (1, 0, 1, 3, 2),
#   5: (1, 0, 1, 1, 0),
#   8: (1, 0, 1, 1, 0),
#   10: (1, 0, 0, 0, 0),
#   14: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 1, 0),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   26: (1, 0, 1, 1, 0),
#   27: (1, 0, 1, 1, 1),
#   28: (1, 0, 1, 1, 1),
#   30: (1, 0, 1, 1, 1),
#   31: (1, 0, 1, 1, 1),
#   35: (1, 0, 0, 0, 0)},
#  (0, 4, 172, 506, 701): {0: (1, 1, 3, 3, 2),
#   4: (1, 0, 1, 3, 2),
#   5: (1, 0, 1, 1, 0),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 0, 0, 0, 0),
#   8: (1, 0, 1, 1, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   17: (1, 0, 0, 0, 0),
#   18: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 1, 0),
#   24: (1, 0, 0, 0, 0),
#   26: (1, 0, 1, 1, 0),
#   27: (1, 0, 1, 1, 1),
#   28: (1, 0, 1, 1, 1),
#   30: (1, 0, 1, 1, 1),
#   31: (1, 0, 1, 1, 1),
#   34: (1, 0, 0, 0, 0)},
#  (0, 4, 172, 506, 722): {0: (1, 3, 4, 3, 2, 4),
#   2: (0, 0, 1, 0, 0, 1),
#   4: (0, 1, 1, 3, 2, 4),
#   5: (0, 1, 1, 1, 0, 1),
#   8: (0, 1, 1, 1, 0, 1),
#   9: (0, 0, 1, 0, 0, 1),
#   21: (0, 1, 1, 1, 0, 1),
#   26: (0, 1, 1, 1, 0, 1),
#   27: (0, 1, 1, 1, 1, 1),
#   28: (0, 1, 1, 1, 1, 1),
#   29: (0, 0, 1, 0, 0, 1),
#   30: (0, 1, 1, 1, 1, 1),
#   31: (0, 1, 1, 1, 1, 1),
#   32: (0, 0, 1, 0, 0, 1),
#   33: (0, 0, 1, 0, 0, 1)},
#  (0, 29, 165, 490, 695): {0: (1, 1, 3, 2, 2),
#   2: (1, 0, 1, 1, 1),
#   3: (1, 0, 0, 0, 0),
#   5: (1, 0, 1, 0, 0),
#   6: (1, 0, 0, 0, 0),
#   9: (1, 0, 1, 1, 1),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   17: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 1, 1),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   26: (1, 0, 1, 1, 1),
#   27: (1, 0, 1, 0, 0),
#   29: (1, 0, 1, 2, 1),
#   30: (1, 0, 1, 0, 0),
#   32: (1, 0, 1, 0, 0),
#   35: (1, 0, 0, 0, 0)},
#  (0, 29, 165, 490, 698): {0: (1, 1, 3, 2, 2),
#   2: (1, 0, 1, 1, 1),
#   5: (1, 0, 1, 0, 0),
#   7: (1, 0, 0, 0, 0),
#   9: (1, 0, 1, 1, 1),
#   10: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   14: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   18: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 1, 1),
#   24: (1, 0, 0, 0, 0),
#   26: (1, 0, 1, 1, 1),
#   27: (1, 0, 1, 0, 0),
#   29: (1, 0, 1, 2, 1),
#   30: (1, 0, 1, 0, 0),
#   32: (1, 0, 1, 0, 0),
#   34: (1, 0, 0, 0, 0)},
#  (0, 29, 165, 490, 722): {0: (1, 4, 3, 2, 2),
#   2: (0, 1, 1, 1, 1),
#   4: (0, 1, 0, 0, 0),
#   5: (0, 1, 1, 0, 0),
#   8: (0, 1, 0, 0, 0),
#   9: (0, 1, 1, 1, 1),
#   21: (0, 1, 1, 1, 1),
#   26: (0, 1, 1, 1, 1),
#   27: (0, 1, 1, 0, 0),
#   28: (0, 1, 0, 0, 0),
#   29: (0, 1, 1, 2, 1),
#   30: (0, 1, 1, 0, 0),
#   31: (0, 1, 0, 0, 0),
#   32: (0, 1, 1, 0, 0),
#   33: (0, 1, 0, 0, 0)},
#  (0, 29, 165, 493, 696): {0: (1, 1, 3, 2, 2),
#   2: (1, 0, 1, 1, 1),
#   3: (1, 0, 0, 0, 0),
#   6: (1, 0, 0, 0, 0),
#   8: (1, 0, 1, 0, 0),
#   9: (1, 0, 1, 1, 1),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   14: (1, 0, 0, 0, 0),
#   18: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 1, 1),
#   22: (1, 0, 0, 0, 0),
#   24: (1, 0, 0, 0, 0),
#   26: (1, 0, 1, 1, 1),
#   28: (1, 0, 1, 0, 0),
#   29: (1, 0, 1, 2, 1),
#   31: (1, 0, 1, 0, 0),
#   33: (1, 0, 1, 0, 0),
#   34: (1, 0, 0, 0, 0)},
#  (0, 29, 165, 493, 699): {0: (1, 1, 3, 2, 2),
#   2: (1, 0, 1, 1, 1),
#   7: (1, 0, 0, 0, 0),
#   8: (1, 0, 1, 0, 0),
#   9: (1, 0, 1, 1, 1),
#   10: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   15: (1, 0, 0, 0, 0),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   21: (1, 0, 1, 1, 1),
#   23: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   26: (1, 0, 1, 1, 1),
#   28: (1, 0, 1, 0, 0),
#   29: (1, 0, 1, 2, 1),
#   31: (1, 0, 1, 0, 0),
#   33: (1, 0, 1, 0, 0),
#   35: (1, 0, 0, 0, 0)},
#  (0, 29, 165, 493, 722): {0: (1, 4, 3, 2, 2),
#   2: (0, 1, 1, 1, 1),
#   4: (0, 1, 0, 0, 0),
#   5: (0, 1, 0, 0, 0),
#   8: (0, 1, 1, 0, 0),
#   9: (0, 1, 1, 1, 1),
#   21: (0, 1, 1, 1, 1),
#   26: (0, 1, 1, 1, 1),
#   27: (0, 1, 0, 0, 0),
#   28: (0, 1, 1, 0, 0),
#   29: (0, 1, 1, 2, 1),
#   30: (0, 1, 0, 0, 0),
#   31: (0, 1, 1, 0, 0),
#   32: (0, 1, 0, 0, 0),
#   33: (0, 1, 1, 0, 0)},
#  (1, 4, 29, 204, 732): {1: (1, 4, 2, 4, 4),
#   3: (1, 1, 1, 1, 1),
#   4: (1, 1, 2, 4, 4),
#   5: (1, 1, 0, 1, 1),
#   7: (1, 1, 1, 1, 1),
#   8: (1, 1, 0, 1, 1),
#   10: (1, 1, 1, 1, 1),
#   13: (1, 1, 1, 1, 1),
#   14: (1, 1, 0, 1, 1),
#   17: (1, 1, 0, 1, 1),
#   29: (1, 1, 0, 4, 1),
#   32: (1, 1, 0, 1, 1),
#   33: (1, 1, 0, 1, 1),
#   34: (1, 1, 0, 1, 1),
#   35: (1, 1, 0, 1, 1)},
#  (1, 4, 29, 238, 732): {1: (1, 2, 4, 2, 4, 4),
#   3: (1, 0, 1, 0, 1, 1),
#   4: (1, 0, 1, 0, 4, 4),
#   5: (1, 0, 1, 0, 1, 1),
#   7: (1, 0, 1, 0, 1, 1),
#   8: (1, 0, 1, 0, 1, 1),
#   10: (1, 0, 1, 0, 1, 1),
#   13: (1, 0, 1, 0, 1, 1),
#   14: (1, 1, 1, 1, 1, 1),
#   17: (1, 1, 1, 1, 1, 1),
#   29: (1, 1, 1, 2, 4, 1),
#   32: (1, 0, 1, 0, 1, 1),
#   33: (1, 0, 1, 0, 1, 1),
#   34: (1, 1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1, 1)},
#  (1, 4, 29, 309, 732): {1: (0, 1, 0, 0, 4, 4, 4),
#   3: (0, 1, 0, 0, 1, 1, 1),
#   4: (1, 1, 2, 2, 1, 4, 4),
#   5: (1, 1, 1, 1, 1, 1, 1),
#   7: (0, 1, 0, 0, 1, 1, 1),
#   8: (1, 1, 1, 1, 1, 1, 1),
#   10: (0, 1, 0, 0, 1, 1, 1),
#   13: (0, 1, 0, 0, 1, 1, 1),
#   14: (0, 1, 0, 0, 1, 1, 1),
#   17: (0, 1, 0, 0, 1, 1, 1),
#   29: (1, 1, 2, 1, 1, 1, 4),
#   32: (1, 1, 1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1, 1, 1),
#   34: (0, 1, 0, 0, 1, 1, 1),
#   35: (0, 1, 0, 0, 1, 1, 1)},
#  (1, 4, 204, 540, 702): {1: (1, 1, 3, 2, 3),
#   2: (1, 0, 0, 0, 0),
#   3: (1, 1, 1, 1, 1),
#   4: (1, 1, 1, 2, 3),
#   5: (1, 1, 1, 0, 1),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 1, 1, 1, 1),
#   8: (1, 1, 1, 0, 1),
#   10: (1, 1, 1, 1, 1),
#   11: (1, 0, 0, 0, 0),
#   13: (1, 1, 1, 1, 1),
#   14: (1, 1, 1, 0, 1),
#   15: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 0, 1),
#   18: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   27: (1, 0, 0, 0, 0),
#   28: (1, 0, 0, 0, 0)},
#  (1, 4, 204, 540, 706): {1: (1, 1, 3, 2, 3),
#   3: (1, 1, 1, 1, 1),
#   4: (1, 1, 1, 2, 3),
#   5: (1, 1, 1, 0, 1),
#   7: (1, 1, 1, 1, 1),
#   8: (1, 1, 1, 0, 1),
#   9: (1, 0, 0, 0, 0),
#   10: (1, 1, 1, 1, 1),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 1, 1, 1, 1),
#   14: (1, 1, 1, 0, 1),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 0, 1),
#   19: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   24: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   26: (1, 0, 0, 0, 0),
#   30: (1, 0, 0, 0, 0),
#   31: (1, 0, 0, 0, 0)},
#  (1, 4, 204, 540, 732): {1: (1, 1, 4, 2, 4, 3, 3),
#   3: (1, 1, 1, 1, 1, 1, 1),
#   4: (1, 1, 1, 2, 4, 3, 1),
#   5: (1, 1, 1, 0, 1, 1, 1),
#   7: (1, 1, 1, 1, 1, 1, 1),
#   8: (1, 1, 1, 0, 1, 1, 1),
#   10: (1, 1, 1, 1, 1, 1, 1),
#   13: (1, 1, 1, 1, 1, 1, 1),
#   14: (1, 1, 1, 0, 1, 1, 1),
#   17: (1, 1, 1, 0, 1, 1, 1),
#   29: (1, 0, 1, 0, 1, 0, 0),
#   32: (1, 0, 1, 0, 1, 0, 0),
#   33: (1, 0, 1, 0, 1, 0, 0),
#   34: (1, 0, 1, 0, 1, 0, 0),
#   35: (1, 0, 1, 0, 1, 0, 0)},
#  (1, 4, 204, 543, 703): {1: (1, 1, 3, 2, 3),
#   2: (1, 0, 0, 0, 0),
#   3: (1, 1, 1, 1, 1),
#   4: (1, 1, 1, 2, 3),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 1, 1, 1, 1),
#   10: (1, 1, 1, 1, 1),
#   11: (1, 0, 0, 0, 0),
#   13: (1, 1, 1, 1, 1),
#   15: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 0, 0, 0),
#   24: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   30: (1, 0, 0, 0, 0),
#   31: (1, 0, 0, 0, 0),
#   32: (1, 1, 1, 0, 1),
#   33: (1, 1, 1, 0, 1),
#   34: (1, 1, 1, 0, 1),
#   35: (1, 1, 1, 0, 1)},
#  (1, 4, 204, 543, 707): {1: (1, 1, 3, 2, 3),
#   3: (1, 1, 1, 1, 1),
#   4: (1, 1, 1, 2, 3),
#   7: (1, 1, 1, 1, 1),
#   9: (1, 0, 0, 0, 0),
#   10: (1, 1, 1, 1, 1),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 1, 1, 1, 1),
#   16: (1, 0, 0, 0, 0),
#   18: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   26: (1, 0, 0, 0, 0),
#   27: (1, 0, 0, 0, 0),
#   28: (1, 0, 0, 0, 0),
#   32: (1, 1, 1, 0, 1),
#   33: (1, 1, 1, 0, 1),
#   34: (1, 1, 1, 0, 1),
#   35: (1, 1, 1, 0, 1)},
#  (1, 4, 204, 543, 732): {1: (1, 1, 4, 2, 4, 3, 3),
#   3: (1, 1, 1, 1, 1, 1, 1),
#   4: (1, 1, 1, 2, 4, 3, 1),
#   5: (1, 0, 1, 0, 1, 0, 0),
#   7: (1, 1, 1, 1, 1, 1, 1),
#   8: (1, 0, 1, 0, 1, 0, 0),
#   10: (1, 1, 1, 1, 1, 1, 1),
#   13: (1, 1, 1, 1, 1, 1, 1),
#   14: (1, 0, 1, 0, 1, 0, 0),
#   17: (1, 0, 1, 0, 1, 0, 0),
#   29: (1, 0, 1, 0, 1, 0, 0),
#   32: (1, 1, 1, 0, 1, 1, 1),
#   33: (1, 1, 1, 0, 1, 1, 1),
#   34: (1, 1, 1, 0, 1, 1, 1),
#   35: (1, 1, 1, 0, 1, 1, 1)},
#  (1, 29, 238, 548, 704): {1: (1, 1, 2, 3, 2),
#   2: (1, 0, 0, 0, 0),
#   3: (1, 1, 0, 1, 0),
#   6: (1, 0, 0, 0, 0),
#   7: (1, 1, 0, 1, 0),
#   8: (1, 1, 0, 1, 0),
#   14: (1, 1, 1, 1, 1),
#   15: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 1, 1),
#   18: (1, 0, 0, 0, 0),
#   19: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   26: (1, 0, 0, 0, 0),
#   27: (1, 0, 0, 0, 0),
#   29: (1, 1, 2, 1, 1),
#   31: (1, 0, 0, 0, 0),
#   33: (1, 1, 0, 1, 0),
#   34: (1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1)},
#  (1, 29, 238, 548, 708): {1: (1, 1, 2, 3, 2),
#   3: (1, 1, 0, 1, 0),
#   7: (1, 1, 0, 1, 0),
#   8: (1, 1, 0, 1, 0),
#   9: (1, 0, 0, 0, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   14: (1, 1, 1, 1, 1),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 1, 1),
#   20: (1, 0, 0, 0, 0),
#   21: (1, 0, 0, 0, 0),
#   24: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   28: (1, 0, 0, 0, 0),
#   29: (1, 1, 2, 1, 1),
#   30: (1, 0, 0, 0, 0),
#   33: (1, 1, 0, 1, 0),
#   34: (1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1)},
#  (1, 29, 238, 548, 732): {1: (1, 1, 4, 3, 2, 2),
#   3: (1, 1, 1, 1, 0, 0),
#   4: (0, 1, 1, 0, 0, 0),
#   5: (0, 1, 1, 0, 0, 0),
#   7: (1, 1, 1, 1, 0, 0),
#   8: (1, 1, 1, 1, 0, 0),
#   10: (0, 1, 1, 0, 0, 0),
#   13: (0, 1, 1, 0, 0, 0),
#   14: (1, 1, 1, 1, 1, 1),
#   17: (1, 1, 1, 1, 1, 1),
#   29: (1, 1, 1, 1, 1, 2),
#   32: (0, 1, 1, 0, 0, 0),
#   33: (1, 1, 1, 1, 0, 0),
#   34: (1, 1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1, 1)},
#  (1, 29, 238, 563, 705): {1: (1, 1, 2, 3, 2),
#   2: (1, 0, 0, 0, 0),
#   5: (1, 1, 0, 1, 0),
#   10: (1, 1, 0, 1, 0),
#   11: (1, 0, 0, 0, 0),
#   12: (1, 0, 0, 0, 0),
#   13: (1, 1, 0, 1, 0),
#   14: (1, 1, 1, 1, 1),
#   16: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 1, 1),
#   18: (1, 0, 0, 0, 0),
#   20: (1, 0, 0, 0, 0),
#   23: (1, 0, 0, 0, 0),
#   26: (1, 0, 0, 0, 0),
#   28: (1, 0, 0, 0, 0),
#   29: (1, 1, 2, 1, 1),
#   30: (1, 0, 0, 0, 0),
#   32: (1, 1, 0, 1, 0),
#   34: (1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1)},
#  (1, 29, 238, 563, 709): {1: (1, 1, 2, 3, 2),
#   5: (1, 1, 0, 1, 0),
#   6: (1, 0, 0, 0, 0),
#   9: (1, 0, 0, 0, 0),
#   10: (1, 1, 0, 1, 0),
#   13: (1, 1, 0, 1, 0),
#   14: (1, 1, 1, 1, 1),
#   15: (1, 0, 0, 0, 0),
#   17: (1, 1, 1, 1, 1),
#   19: (1, 0, 0, 0, 0),
#   21: (1, 0, 0, 0, 0),
#   22: (1, 0, 0, 0, 0),
#   24: (1, 0, 0, 0, 0),
#   25: (1, 0, 0, 0, 0),
#   27: (1, 0, 0, 0, 0),
#   29: (1, 1, 2, 1, 1),
#   31: (1, 0, 0, 0, 0),
#   32: (1, 1, 0, 1, 0),
#   34: (1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1)},
#  (1, 29, 238, 563, 732): {1: (1, 1, 4, 3, 2, 2),
#   3: (0, 1, 1, 0, 0, 0),
#   4: (0, 1, 1, 0, 0, 0),
#   5: (1, 1, 1, 1, 0, 0),
#   7: (0, 1, 1, 0, 0, 0),
#   8: (0, 1, 1, 0, 0, 0),
#   10: (1, 1, 1, 1, 0, 0),
#   13: (1, 1, 1, 1, 0, 0),
#   14: (1, 1, 1, 1, 1, 1),
#   17: (1, 1, 1, 1, 1, 1),
#   29: (1, 1, 1, 1, 1, 2),
#   32: (1, 1, 1, 1, 0, 0),
#   33: (0, 1, 1, 0, 0, 0),
#   34: (1, 1, 1, 1, 1, 1),
#   35: (1, 1, 1, 1, 1, 1)},
#  (4, 29, 309, 613, 710): {2: (0, 1, 0, 0, 0),
#   3: (1, 1, 0, 0, 0),
#   4: (1, 1, 1, 2, 2),
#   5: (1, 1, 1, 1, 1),
#   6: (0, 1, 0, 0, 0),
#   8: (1, 1, 1, 1, 1),
#   10: (1, 1, 0, 0, 0),
#   11: (0, 1, 0, 0, 0),
#   14: (1, 1, 0, 0, 0),
#   16: (0, 1, 0, 0, 0),
#   18: (0, 1, 0, 0, 0),
#   22: (0, 1, 0, 0, 0),
#   25: (0, 1, 0, 0, 0),
#   26: (0, 1, 0, 0, 0),
#   29: (1, 1, 1, 2, 1),
#   30: (0, 1, 0, 0, 0),
#   31: (0, 1, 0, 0, 0),
#   32: (1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1),
#   34: (1, 1, 0, 0, 0)},
#  (4, 29, 309, 613, 712): {3: (1, 1, 0, 0, 0),
#   4: (1, 1, 1, 2, 2),
#   5: (1, 1, 1, 1, 1),
#   8: (1, 1, 1, 1, 1),
#   9: (0, 1, 0, 0, 0),
#   10: (1, 1, 0, 0, 0),
#   12: (0, 1, 0, 0, 0),
#   14: (1, 1, 0, 0, 0),
#   15: (0, 1, 0, 0, 0),
#   19: (0, 1, 0, 0, 0),
#   20: (0, 1, 0, 0, 0),
#   21: (0, 1, 0, 0, 0),
#   23: (0, 1, 0, 0, 0),
#   24: (0, 1, 0, 0, 0),
#   27: (0, 1, 0, 0, 0),
#   28: (0, 1, 0, 0, 0),
#   29: (1, 1, 1, 2, 1),
#   32: (1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1),
#   34: (1, 1, 0, 0, 0)},
#  (4, 29, 309, 613, 732): {1: (1, 0, 0, 0, 0),
#   3: (1, 1, 0, 0, 0),
#   4: (1, 1, 1, 2, 2),
#   5: (1, 1, 1, 1, 1),
#   7: (1, 0, 0, 0, 0),
#   8: (1, 1, 1, 1, 1),
#   10: (1, 1, 0, 0, 0),
#   13: (1, 0, 0, 0, 0),
#   14: (1, 1, 0, 0, 0),
#   17: (1, 0, 0, 0, 0),
#   29: (1, 1, 1, 2, 1),
#   32: (1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1),
#   34: (1, 1, 0, 0, 0),
#   35: (1, 0, 0, 0, 0)},
#  (4, 29, 309, 636, 711): {2: (0, 1, 0, 0, 0),
#   4: (1, 1, 1, 2, 2),
#   5: (1, 1, 1, 1, 1),
#   7: (1, 1, 0, 0, 0),
#   8: (1, 1, 1, 1, 1),
#   12: (0, 1, 0, 0, 0),
#   13: (1, 1, 0, 0, 0),
#   15: (0, 1, 0, 0, 0),
#   17: (1, 1, 0, 0, 0),
#   19: (0, 1, 0, 0, 0),
#   20: (0, 1, 0, 0, 0),
#   23: (0, 1, 0, 0, 0),
#   24: (0, 1, 0, 0, 0),
#   26: (0, 1, 0, 0, 0),
#   29: (1, 1, 1, 2, 1),
#   30: (0, 1, 0, 0, 0),
#   31: (0, 1, 0, 0, 0),
#   32: (1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1),
#   35: (1, 1, 0, 0, 0)},
#  (4, 29, 309, 636, 713): {4: (1, 1, 1, 2, 2),
#   5: (1, 1, 1, 1, 1),
#   6: (0, 1, 0, 0, 0),
#   7: (1, 1, 0, 0, 0),
#   8: (1, 1, 1, 1, 1),
#   9: (0, 1, 0, 0, 0),
#   11: (0, 1, 0, 0, 0),
#   13: (1, 1, 0, 0, 0),
#   16: (0, 1, 0, 0, 0),
#   17: (1, 1, 0, 0, 0),
#   18: (0, 1, 0, 0, 0),
#   21: (0, 1, 0, 0, 0),
#   22: (0, 1, 0, 0, 0),
#   25: (0, 1, 0, 0, 0),
#   27: (0, 1, 0, 0, 0),
#   28: (0, 1, 0, 0, 0),
#   29: (1, 1, 1, 2, 1),
#   32: (1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1),
#   35: (1, 1, 0, 0, 0)},
#  (4, 29, 309, 636, 732): {1: (1, 0, 0, 0, 0),
#   3: (1, 0, 0, 0, 0),
#   4: (1, 1, 1, 2, 2),
#   5: (1, 1, 1, 1, 1),
#   7: (1, 1, 0, 0, 0),
#   8: (1, 1, 1, 1, 1),
#   10: (1, 0, 0, 0, 0),
#   13: (1, 1, 0, 0, 0),
#   14: (1, 0, 0, 0, 0),
#   17: (1, 1, 0, 0, 0),
#   29: (1, 1, 1, 2, 1),
#   32: (1, 1, 1, 1, 1),
#   33: (1, 1, 1, 1, 1),
#   34: (1, 0, 0, 0, 0),
#   35: (1, 1, 0, 0, 0)}}



rootValuesaa2a3a4Y34 = dict()
for u in raysFibersInaa2a3a4_lower_dim.keys():
    l = len(raysFibersInaa2a3a4_lower_dim[u])
    rootValuesaa2a3a4Y34[u] = {r: tuple([raysFibersInaa2a3a4_lower_dim[u][k][r] for k in range(0,l)]) for r in rootsForConesaa2a3a4SummandY34[u]}


rootValuesaa2a3a4Y8 = dict()
for u in raysFibersInaa2a3a4_lower_dim.keys():
    l = len(raysFibersInaa2a3a4_lower_dim[u])
    rootValuesaa2a3a4Y8[u] = {r: tuple([raysFibersInaa2a3a4_lower_dim[u][k][r] for k in range(0,l)]) for r in rootsForConesaa2a3a4SummandY8[u]}


# We record the data:
# for u in rootValuesaa2a3a4Y34.keys():
#     print str(u) + ': ' + str(rootValuesaa2a3a4Y34[u])
# 
# (4, 29, 309, 613, 732): {17: (1, 0, 0, 0, 0), 10: (1, 1, 0, 0, 0), 33: (1, 1, 1, 1, 1)}
# (0, 1, 4, 204, 717): {24: (0, 0, 0, 1, 0), 10: (1, 0, 0, 1, 0), 27: (0, 0, 0, 1, 0)}
# (1, 29, 238, 563, 732): {17: (1, 1, 1, 1, 1, 1), 10: (1, 1, 1, 1, 0, 0), 33: (0, 1, 1, 0, 0, 0)}
# (1, 4, 29, 309, 732): {17: (0, 1, 0, 0, 1, 1, 1), 10: (0, 1, 0, 0, 1, 1, 1), 33: (1, 1, 1, 1, 1, 1, 1)}
# (1, 29, 238, 548, 732): {17: (1, 1, 1, 1, 1, 1), 10: (0, 1, 1, 0, 0, 0), 33: (1, 1, 1, 1, 0, 0)}
# (0, 1, 29, 238, 716): {24: (0, 0, 1, 0, 0, 1), 17: (1, 1, 1, 0, 0, 1), 26: (0, 0, 1, 0, 0, 1)}
# (0, 1, 29, 165, 493): {33: (0, 0, 0, 1, 0), 26: (0, 1, 1, 1, 0)}
# (0, 4, 172, 487, 722): {33: (0, 1, 1, 1, 0, 1), 26: (0, 0, 1, 0, 0, 1), 27: (0, 1, 1, 1, 1, 1)}
# (0, 4, 172, 506, 700): {10: (1, 0, 0, 0, 0), 27: (1, 0, 1, 1, 1), 26: (1, 0, 1, 1, 0)}
# (0, 1, 4, 172, 506): {26: (0, 0, 0, 0, 1, 1), 27: (1, 0, 0, 0, 1, 1)}
# (0, 1, 4, 158, 473): {24: (0, 1, 0, 1, 0), 17: (0, 0, 0, 1, 0)}
# (0, 4, 29, 309, 613): {33: (1, 1, 1, 1, 0), 10: (0, 0, 0, 1, 0)}
# (0, 1, 158, 473, 716): {24: (0, 1, 0, 1, 1), 17: (0, 0, 0, 1, 1), 26: (0, 0, 0, 1, 0)}
# (0, 1, 158, 482, 716): {24: (0, 1, 0, 1, 1), 17: (0, 0, 0, 1, 0), 26: (0, 0, 0, 1, 1)}
# (0, 4, 29, 309, 722): {33: (1, 1, 1, 1, 1, 1, 0), 26: (0, 0, 0, 1, 1, 1, 0), 27: (0, 0, 0, 1, 1, 1, 0)}
# (1, 4, 204, 540, 732): {17: (1, 1, 1, 0, 1, 1, 1), 10: (1, 1, 1, 1, 1, 1, 1), 33: (1, 0, 1, 0, 1, 0, 0)}
# (0, 1, 4, 172, 717): {24: (0, 0, 0, 1, 0), 10: (0, 0, 0, 1, 0), 27: (0, 1, 0, 1, 0)}
# (1, 29, 238, 563, 709): {24: (1, 0, 0, 0, 0), 17: (1, 1, 1, 1, 1), 10: (1, 1, 0, 1, 0), 27: (1, 0, 0, 0, 0)}
# (0, 1, 29, 158, 716): {24: (0, 0, 1, 1, 1), 17: (0, 0, 1, 1, 0), 26: (0, 0, 1, 1, 0)}
# (0, 1, 29, 238, 563): {17: (1, 1, 0, 0, 1, 1), 10: (0, 0, 0, 0, 1, 1)}
# (0, 1, 4, 158, 717): {24: (0, 0, 1, 1, 0), 10: (0, 0, 0, 1, 0), 27: (0, 0, 0, 1, 0)}
# (0, 4, 29, 165, 722): {33: (0, 0, 1, 0, 1, 1), 26: (0, 1, 1, 1, 1, 1), 27: (0, 0, 1, 0, 1, 1)}
# (0, 1, 4, 158, 691): {24: (1, 0, 1, 0, 0), 33: (1, 0, 0, 0, 0)}
# (0, 1, 4, 172, 692): {17: (1, 0, 0, 0, 0), 27: (1, 0, 1, 0, 0)}
# (0, 4, 29, 309, 636): {17: (0, 0, 0, 1, 0), 33: (1, 1, 1, 1, 0)}
# (0, 1, 4, 204, 687): {10: (1, 1, 0, 0, 0), 26: (1, 0, 0, 0, 0)}
# (0, 4, 29, 309, 691): {24: (0, 1, 0, 0, 0), 33: (0, 1, 1, 1, 1)}
# (0, 1, 29, 165, 490): {26: (0, 1, 1, 1, 0), 27: (0, 0, 0, 1, 0)}
# (1, 29, 238, 563, 705): {17: (1, 1, 1, 1, 1), 10: (1, 1, 0, 1, 0), 26: (1, 0, 0, 0, 0)}
# (1, 4, 204, 543, 703): {24: (1, 0, 0, 0, 0), 33: (1, 1, 1, 0, 1), 10: (1, 1, 1, 1, 1)}
# (0, 29, 165, 493, 699): {17: (1, 0, 0, 0, 0), 10: (1, 0, 0, 0, 0), 26: (1, 0, 1, 1, 1), 33: (1, 0, 1, 0, 0)}
# (0, 29, 165, 490, 695): {17: (1, 0, 0, 0, 0), 26: (1, 0, 1, 1, 1), 27: (1, 0, 1, 0, 0)}
# (0, 1, 29, 165, 716): {24: (0, 0, 0, 0, 1, 1), 17: (0, 0, 0, 0, 1, 1), 26: (0, 1, 1, 0, 1, 1)}
# (0, 29, 165, 493, 696): {24: (1, 0, 0, 0, 0), 33: (1, 0, 1, 0, 0), 26: (1, 0, 1, 1, 1)}
# (0, 1, 4, 204, 543): {33: (0, 0, 0, 0, 1, 1, 1), 10: (1, 0, 0, 0, 1, 1, 1)}
# (0, 1, 4, 158, 482): {24: (0, 1, 0, 1, 0), 26: (0, 0, 0, 1, 0)}
# (0, 1, 158, 473, 689): {24: (1, 0, 1, 0, 1), 17: (1, 0, 1, 0, 0), 10: (1, 0, 0, 0, 0), 33: (1, 0, 0, 0, 0)}
# (0, 29, 165, 490, 698): {24: (1, 0, 0, 0, 0), 10: (1, 0, 0, 0, 0), 27: (1, 0, 1, 0, 0), 26: (1, 0, 1, 1, 1)}
# (0, 4, 172, 506, 701): {24: (1, 0, 0, 0, 0), 17: (1, 0, 0, 0, 0), 26: (1, 0, 1, 1, 0), 27: (1, 0, 1, 1, 1)}
# (0, 29, 165, 490, 722): {33: (0, 1, 0, 0, 0), 26: (0, 1, 1, 1, 1), 27: (0, 1, 1, 0, 0)}
# (0, 29, 165, 493, 722): {33: (0, 1, 1, 0, 0), 26: (0, 1, 1, 1, 1), 27: (0, 1, 0, 0, 0)}
# (4, 29, 309, 613, 712): {24: (0, 1, 0, 0, 0), 33: (1, 1, 1, 1, 1), 10: (1, 1, 0, 0, 0), 27: (0, 1, 0, 0, 0)}
# (0, 4, 172, 487, 694): {17: (1, 0, 0, 0, 0), 10: (1, 0, 0, 0, 0), 27: (1, 0, 1, 1, 1), 33: (1, 0, 1, 1, 0)}
# (1, 29, 238, 548, 708): {24: (1, 0, 0, 0, 0), 17: (1, 1, 1, 1, 1), 33: (1, 1, 0, 1, 0)}
# (0, 4, 29, 172, 722): {33: (0, 1, 1, 0, 1), 26: (0, 1, 1, 0, 1), 27: (0, 1, 1, 1, 1)}
# (1, 4, 204, 543, 732): {17: (1, 0, 1, 0, 1, 0, 0), 10: (1, 1, 1, 1, 1, 1, 1), 33: (1, 1, 1, 0, 1, 1, 1)}
# (0, 1, 29, 165, 687): {10: (1, 0, 0, 0, 0), 26: (1, 0, 0, 1, 1)}
# (1, 4, 204, 543, 707): {33: (1, 1, 1, 0, 1), 10: (1, 1, 1, 1, 1), 27: (1, 0, 0, 0, 0), 26: (1, 0, 0, 0, 0)}
# (0, 1, 158, 482, 690): {24: (1, 0, 1, 0, 1), 33: (1, 0, 0, 0, 0), 26: (1, 0, 1, 0, 0), 27: (1, 0, 0, 0, 0)}
# (0, 1, 4, 172, 487): {33: (0, 0, 0, 0, 1, 1), 27: (1, 0, 0, 0, 1, 1)}
# (4, 29, 309, 613, 710): {33: (1, 1, 1, 1, 1), 10: (1, 1, 0, 0, 0), 26: (0, 1, 0, 0, 0)}
# (4, 29, 309, 636, 711): {24: (0, 1, 0, 0, 0), 17: (1, 1, 0, 0, 0), 26: (0, 1, 0, 0, 0), 33: (1, 1, 1, 1, 1)}
# (0, 1, 29, 238, 692): {17: (1, 1, 1, 0, 0), 27: (1, 0, 0, 0, 0)}
# (4, 29, 309, 636, 713): {17: (1, 1, 0, 0, 0), 27: (0, 1, 0, 0, 0), 33: (1, 1, 1, 1, 1)}
# (0, 4, 172, 487, 697): {24: (1, 0, 0, 0, 0), 33: (1, 0, 1, 1, 0), 27: (1, 0, 1, 1, 1)}
# (0, 4, 172, 506, 722): {33: (0, 0, 1, 0, 0, 1), 26: (0, 1, 1, 1, 0, 1), 27: (0, 1, 1, 1, 1, 1)}
# (0, 1, 4, 204, 540): {17: (0, 0, 0, 0, 1, 1, 1), 10: (1, 0, 0, 0, 1, 1, 1)}
# (1, 4, 29, 204, 732): {17: (1, 1, 0, 1, 1), 10: (1, 1, 1, 1, 1), 33: (1, 1, 0, 1, 1)}
# (1, 4, 204, 540, 702): {17: (1, 1, 1, 0, 1), 10: (1, 1, 1, 1, 1), 27: (1, 0, 0, 0, 0)}
# (0, 1, 158, 482, 693): {24: (1, 0, 1, 0, 1), 10: (1, 0, 0, 0, 0), 26: (1, 0, 1, 0, 0)}
# (0, 1, 29, 238, 548): {17: (1, 1, 0, 0, 1, 1), 33: (0, 0, 0, 0, 1, 1)}
# (1, 4, 29, 238, 732): {17: (1, 1, 1, 1, 1, 1), 10: (1, 0, 1, 0, 1, 1), 33: (1, 0, 1, 0, 1, 1)}
# (1, 29, 238, 548, 704): {17: (1, 1, 1, 1, 1), 26: (1, 0, 0, 0, 0), 27: (1, 0, 0, 0, 0), 33: (1, 1, 0, 1, 0)}
# (4, 29, 309, 636, 732): {17: (1, 1, 0, 0, 0), 10: (1, 0, 0, 0, 0), 33: (1, 1, 1, 1, 1)}
# (1, 4, 204, 540, 706): {24: (1, 0, 0, 0, 0), 17: (1, 1, 1, 0, 1), 10: (1, 1, 1, 1, 1), 26: (1, 0, 0, 0, 0)}
# (0, 1, 158, 473, 688): {24: (1, 0, 1, 0, 1), 17: (1, 0, 1, 0, 0), 27: (1, 0, 0, 0, 0)}

# We record the data:
# for u in rootValuesaa2a3a4Y8.keys():
#     print str(u) + ': ' + str(rootValuesaa2a3a4Y8[u])
# 
# (4, 29, 309, 613, 732): {34: (1, 1, 0, 0, 0), 5: (1, 1, 1, 1, 1), 7: (1, 0, 0, 0, 0)}
# (0, 1, 4, 204, 717): {31: (0, 0, 0, 1, 0), 23: (0, 0, 0, 1, 0), 7: (1, 0, 0, 1, 0)}
# (1, 29, 238, 563, 732): {34: (1, 1, 1, 1, 1, 1), 5: (1, 1, 1, 1, 0, 0), 7: (0, 1, 1, 0, 0, 0)}
# (1, 4, 29, 309, 732): {34: (0, 1, 0, 0, 1, 1, 1), 5: (1, 1, 1, 1, 1, 1, 1), 7: (0, 1, 0, 0, 1, 1, 1)}
# (1, 29, 238, 548, 732): {34: (1, 1, 1, 1, 1, 1), 5: (0, 1, 1, 0, 0, 0), 7: (1, 1, 1, 1, 0, 0)}
# (0, 1, 29, 238, 716): {9: (0, 0, 1, 0, 0, 1), 34: (1, 1, 1, 0, 0, 1), 23: (0, 0, 1, 0, 0, 1)}
# (0, 1, 29, 165, 493): {9: (0, 1, 1, 1, 0), 31: (0, 0, 0, 1, 0)}
# (0, 4, 172, 487, 722): {9: (0, 1, 1, 1, 0, 1), 5: (0, 0, 1, 0, 0, 1), 31: (0, 1, 1, 1, 1, 1)}
# (0, 4, 172, 506, 700): {31: (1, 0, 1, 1, 1), 5: (1, 0, 1, 1, 0), 23: (1, 0, 0, 0, 0)}
# (0, 1, 4, 172, 506): {5: (0, 0, 0, 0, 1, 1), 31: (1, 0, 0, 0, 1, 1)}
# (0, 1, 4, 158, 473): {9: (0, 0, 0, 1, 0), 23: (0, 1, 0, 1, 0)}
# (0, 4, 29, 309, 613): {34: (0, 0, 0, 1, 0), 5: (1, 1, 1, 1, 0)}
# (0, 1, 158, 473, 716): {9: (0, 0, 0, 1, 1), 34: (0, 0, 0, 1, 0), 23: (0, 1, 0, 1, 1)}
# (0, 1, 158, 482, 716): {9: (0, 0, 0, 1, 0), 34: (0, 0, 0, 1, 1), 23: (0, 1, 0, 1, 1)}
# (0, 4, 29, 309, 722): {9: (0, 0, 0, 1, 1, 1, 0), 5: (1, 1, 1, 1, 1, 1, 0), 31: (0, 0, 0, 1, 1, 1, 0)}
# (1, 4, 204, 540, 732): {34: (1, 0, 1, 0, 1, 0, 0), 5: (1, 1, 1, 0, 1, 1, 1), 7: (1, 1, 1, 1, 1, 1, 1)}
# (0, 1, 4, 172, 717): {31: (0, 1, 0, 1, 0), 23: (0, 0, 0, 1, 0), 7: (0, 0, 0, 1, 0)}
# (1, 29, 238, 563, 709): {9: (1, 0, 0, 0, 0), 34: (1, 1, 1, 1, 1), 5: (1, 1, 0, 1, 0), 31: (1, 0, 0, 0, 0)}
# (0, 1, 29, 158, 716): {9: (0, 0, 1, 1, 0), 34: (0, 0, 1, 1, 0), 23: (0, 0, 1, 1, 1)}
# (0, 1, 29, 238, 563): {34: (1, 1, 0, 0, 1, 1), 5: (0, 0, 0, 0, 1, 1)}
# (0, 1, 4, 158, 717): {31: (0, 0, 0, 1, 0), 23: (0, 0, 1, 1, 0), 7: (0, 0, 0, 1, 0)}
# (0, 4, 29, 165, 722): {9: (0, 1, 1, 1, 1, 1), 5: (0, 0, 1, 0, 1, 1), 31: (0, 0, 1, 0, 1, 1)}
# (0, 1, 4, 158, 691): {5: (1, 0, 0, 0, 0), 23: (1, 0, 1, 0, 0)}
# (0, 1, 4, 172, 692): {34: (1, 0, 0, 0, 0), 31: (1, 0, 1, 0, 0)}
# (0, 4, 29, 309, 636): {5: (1, 1, 1, 1, 0), 7: (0, 0, 0, 1, 0)}
# (0, 1, 4, 204, 687): {9: (1, 0, 0, 0, 0), 7: (1, 1, 0, 0, 0)}
# (0, 4, 29, 309, 691): {5: (0, 1, 1, 1, 1), 23: (0, 1, 0, 0, 0)}
# (0, 1, 29, 165, 490): {9: (0, 1, 1, 1, 0), 5: (0, 0, 0, 1, 0)}
# (1, 29, 238, 563, 705): {34: (1, 1, 1, 1, 1), 5: (1, 1, 0, 1, 0), 23: (1, 0, 0, 0, 0)}
# (1, 4, 204, 543, 703): {34: (1, 1, 1, 0, 1), 31: (1, 0, 0, 0, 0), 7: (1, 1, 1, 1, 1)}
# (0, 29, 165, 493, 699): {31: (1, 0, 1, 0, 0), 9: (1, 0, 1, 1, 1), 23: (1, 0, 0, 0, 0), 7: (1, 0, 0, 0, 0)}
# (0, 29, 165, 490, 695): {9: (1, 0, 1, 1, 1), 5: (1, 0, 1, 0, 0), 23: (1, 0, 0, 0, 0)}
# (0, 1, 29, 165, 716): {9: (0, 1, 1, 0, 1, 1), 34: (0, 0, 0, 0, 1, 1), 23: (0, 0, 0, 0, 1, 1)}
# (0, 29, 165, 493, 696): {9: (1, 0, 1, 1, 1), 34: (1, 0, 0, 0, 0), 31: (1, 0, 1, 0, 0)}
# (0, 1, 4, 204, 543): {34: (0, 0, 0, 0, 1, 1, 1), 7: (1, 0, 0, 0, 1, 1, 1)}
# (0, 1, 4, 158, 482): {34: (0, 0, 0, 1, 0), 23: (0, 1, 0, 1, 0)}
# (0, 1, 158, 473, 689): {9: (1, 0, 1, 0, 0), 31: (1, 0, 0, 0, 0), 5: (1, 0, 0, 0, 0), 23: (1, 0, 1, 0, 1)}
# (0, 29, 165, 490, 698): {9: (1, 0, 1, 1, 1), 34: (1, 0, 0, 0, 0), 5: (1, 0, 1, 0, 0), 7: (1, 0, 0, 0, 0)}
# (0, 4, 172, 506, 701): {34: (1, 0, 0, 0, 0), 31: (1, 0, 1, 1, 1), 5: (1, 0, 1, 1, 0), 7: (1, 0, 0, 0, 0)}
# (0, 29, 165, 490, 722): {9: (0, 1, 1, 1, 1), 5: (0, 1, 1, 0, 0), 31: (0, 1, 0, 0, 0)}
# (0, 29, 165, 493, 722): {9: (0, 1, 1, 1, 1), 5: (0, 1, 0, 0, 0), 31: (0, 1, 1, 0, 0)}
# (4, 29, 309, 613, 712): {9: (0, 1, 0, 0, 0), 34: (1, 1, 0, 0, 0), 5: (1, 1, 1, 1, 1), 23: (0, 1, 0, 0, 0)}
# (0, 4, 172, 487, 694): {9: (1, 0, 1, 1, 0), 34: (1, 0, 0, 0, 0), 31: (1, 0, 1, 1, 1), 23: (1, 0, 0, 0, 0)}
# (1, 29, 238, 548, 708): {9: (1, 0, 0, 0, 0), 34: (1, 1, 1, 1, 1), 7: (1, 1, 0, 1, 0)}
# (0, 4, 29, 172, 722): {9: (0, 1, 1, 0, 1), 5: (0, 1, 1, 0, 1), 31: (0, 1, 1, 1, 1)}
# (1, 4, 204, 543, 732): {34: (1, 1, 1, 0, 1, 1, 1), 5: (1, 0, 1, 0, 1, 0, 0), 7: (1, 1, 1, 1, 1, 1, 1)}
# (0, 1, 29, 165, 687): {9: (1, 0, 0, 1, 1), 7: (1, 0, 0, 0, 0)}
# (1, 4, 204, 543, 707): {9: (1, 0, 0, 0, 0), 34: (1, 1, 1, 0, 1), 23: (1, 0, 0, 0, 0), 7: (1, 1, 1, 1, 1)}
# (0, 1, 158, 482, 690): {34: (1, 0, 1, 0, 0), 23: (1, 0, 1, 0, 1), 5: (1, 0, 0, 0, 0), 7: (1, 0, 0, 0, 0)}
# (0, 1, 4, 172, 487): {9: (0, 0, 0, 0, 1, 1), 31: (1, 0, 0, 0, 1, 1)}
# (4, 29, 309, 613, 710): {34: (1, 1, 0, 0, 0), 5: (1, 1, 1, 1, 1), 31: (0, 1, 0, 0, 0)}
# (4, 29, 309, 636, 711): {31: (0, 1, 0, 0, 0), 23: (0, 1, 0, 0, 0), 5: (1, 1, 1, 1, 1), 7: (1, 1, 0, 0, 0)}
# (0, 1, 29, 238, 692): {34: (1, 1, 1, 0, 0), 31: (1, 0, 0, 0, 0)}
# (4, 29, 309, 636, 713): {9: (0, 1, 0, 0, 0), 5: (1, 1, 1, 1, 1), 7: (1, 1, 0, 0, 0)}
# (0, 4, 172, 487, 697): {9: (1, 0, 1, 1, 0), 31: (1, 0, 1, 1, 1), 7: (1, 0, 0, 0, 0)}
# (0, 4, 172, 506, 722): {9: (0, 0, 1, 0, 0, 1), 5: (0, 1, 1, 1, 0, 1), 31: (0, 1, 1, 1, 1, 1)}
# (0, 1, 4, 204, 540): {5: (0, 0, 0, 0, 1, 1, 1), 7: (1, 0, 0, 0, 1, 1, 1)}
# (1, 4, 29, 204, 732): {34: (1, 1, 0, 1, 1), 5: (1, 1, 0, 1, 1), 7: (1, 1, 1, 1, 1)}
# (1, 4, 204, 540, 702): {23: (1, 0, 0, 0, 0), 5: (1, 1, 1, 0, 1), 7: (1, 1, 1, 1, 1)}
# (0, 1, 158, 482, 693): {34: (1, 0, 1, 0, 0), 31: (1, 0, 0, 0, 0), 23: (1, 0, 1, 0, 1)}
# (0, 1, 29, 238, 548): {34: (1, 1, 0, 0, 1, 1), 7: (0, 0, 0, 0, 1, 1)}
# (1, 4, 29, 238, 732): {34: (1, 1, 1, 1, 1, 1), 5: (1, 0, 1, 0, 1, 1), 7: (1, 0, 1, 0, 1, 1)}
# (1, 29, 238, 548, 704): {31: (1, 0, 0, 0, 0), 34: (1, 1, 1, 1, 1), 23: (1, 0, 0, 0, 0), 7: (1, 1, 0, 1, 0)}
# (4, 29, 309, 636, 732): {34: (1, 0, 0, 0, 0), 5: (1, 1, 1, 1, 1), 7: (1, 1, 0, 0, 0)}
# (1, 4, 204, 540, 706): {9: (1, 0, 0, 0, 0), 31: (1, 0, 0, 0, 0), 5: (1, 1, 1, 0, 1), 7: (1, 1, 1, 1, 1)}
# (0, 1, 158, 473, 688): {9: (1, 0, 1, 0, 0), 23: (1, 0, 1, 0, 1), 7: (1, 0, 0, 0, 0)}



rootsForConesaa2a3a4SummandY34 = dict()
rootsForConesaa2a3a4SummandY8 = dict()
for u in raysFibersInaa2a3a4_lower_dim.keys():
    rootsForConesaa2a3a4SummandY34[u] = [x for x in rootsForConesaa2a3a4[u] if x in [24, 10, 17, 27, 26, 33]]
    rootsForConesaa2a3a4SummandY8[u] = [x for x in rootsForConesaa2a3a4[u] if x in [9, 5, 7, 23, 31, 34]]



##########################################################
# Analysis for cones with a fixed list of relevant roots #
##########################################################

# There 66 cones to analyze come in various types, according to their associated relevant roots. 

# In order to determine the value of LT(SumY34) and LT(SumY8), we compute the entries of a typical element in each of the 66 cones. We use scalars p0,...,p7 and rewrite the zs in terms of the ps. We then use the inequalities in zs arising from each 'coefficientsImagesInaa2a3a4_lower_dim[u]' to determine inequalities between the non-negative scalars p0,...,p7. This will allow us to compare valuations of the 36 roots to simplify the expressions of their leading terms.

ps = [var("p%s"%i) for i in range(0,7)]

# The dictionary 'dictIneqpsRoots_lower_dim' produced below records the combinatorial and H-representation of the cones in the fiber of the relative interior of the (aa2a3a4) cone. Moreprecisely, for each 5-tuple corresponding to a Bergman cone, containing the corresponding cone C in the fiber we record:
# 1) the number of rays on the closure of C
# 2) dictz = the correspondence between the scalars zs of the rays in the Bergman cone and the scalars ps of the rays of C
# 3) ineqP = the linear combinations involving the non-negative scalars ps that must be STRICTLY positive 
# 4) rootsV = the vector in R^36/R.1 corresponding to a typical vector in C.

dictIneqpsRoots_lower_dim = dict()
for u in aa2a3a4_fine_overlaps_lower_dim.keys():
    print u
    raysP= [vector(x) for x in Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[u]).rays()]
    paramV = sum([ps[i]*vector(raysP[i]) for i in range(0,len(raysP))])
    dictz = {zs[j]: paramV[j] for j in range(0,len(zs))}    
    rootsV = vector([dictz[zs[i]] for i in range(0,len(zs))])*matrix(cone_from_flat_numbers(list(u)))
    ineqP = tuple([SR(coefficientsImagesInaa2a3a4_lower_dim[u][j]).substitute(dictz) for j in range(0,len(coefficientsImagesInaa2a3a4_lower_dim[u]))])
    dictIneqpsRoots_lower_dim[u] = (len(raysP),dictz, ineqP,rootsV)

# # We record the output
# print dictIneqpsRoots_lower_dim
# {(4, 29, 309, 613, 732): (5, {z5: p0, z4: p1, z3: p2 + p3 + p4, z2: p3, z1: p3 + p4}, (3*p0 + 2*p1, p2, p4, p3), (0, p0, 0, p0 + p1, p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p1 + p2 + p3 + p4, 0, p0, p0 + p1 + p2 + p3 + p4, 0, p0 + p1, 0, 0, p0, p0 + p1, 0, 0, p0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + p2 + 2*p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4, p0 + p1, p0)), (0, 1, 4, 204, 717): (5, {z5: p3, z4: p0, z3: p0 + p2 + 3*p3, z2: p0 + p1 + p2 + 3*p3, z1: p1 + p2 + 3*p3 + p4}, (p4, p1, p2, p0 + 3*p3), (p1 + p2 + 4*p3 + p4, 2*p0 + p1 + p2 + 4*p3, 0, p0 + p3, 2*p0 + p2 + 4*p3, 0, 0, p0 + p3, 0, 0, p0 + p3, 0, 0, p0 + p3, 0, 0, 0, 0, p3, 0, 0, 0, 0, p3, p3, p3, 0, p3, p3, 0, p3, p3, 0, 0, 0, 0)), (1, 29, 238, 563, 732): (6, {z5: p1 + p2, z4: p0 + p3, z3: p4 + p5, z2: p5, z1: 3*p2 + 2*p3 + p4 + p5}, (2*p0 + 3*p1, 3*p2 + 2*p3, p4, p5), (0, p0 + p1 + 4*p2 + 3*p3 + 2*p4 + 2*p5, 0, p1 + p2, p1 + p2, p0 + p1 + p2 + p3, 0, p1 + p2, p1 + p2, 0, p0 + p1 + p2 + p3, 0, 0, p0 + p1 + p2 + p3, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + p2 + p3 + p4 + 2*p5, 0, 0, p0 + p1 + p2 + p3, p1 + p2, p0 + p1 + p2 + p3 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5)), (1, 4, 29, 309, 732): (7, {z5: p1 + p4 + p5 + p6, z4: p0 + p2 + p3, z3: p2 + 3*p6, z2: p2 + p3 + 3*p5 + 3*p6, z1: 3*p4 + 3*p5 + 3*p6}, (3*p1, p0 + 3*p4, p3 + 3*p5, p2 + 3*p6), (0, p1 + 4*p4 + 4*p5 + 4*p6, 0, p1 + p4 + p5 + p6, p0 + p1 + 2*p2 + 2*p3 + p4 + 4*p5 + 4*p6, p0 + p1 + p2 + p3 + p4 + p5 + p6, 0, p1 + p4 + p5 + p6, p0 + p1 + p2 + p3 + p4 + p5 + p6, 0, p1 + p4 + p5 + p6, 0, 0, p1 + p4 + p5 + p6, p1 + p4 + p5 + p6, 0, 0, p1 + p4 + p5 + p6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + 2*p2 + p3 + p4 + p5 + 4*p6, 0, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p1 + p2 + p3 + p4 + p5 + p6, p1 + p4 + p5 + p6, p1 + p4 + p5 + p6)), (1, 29, 238, 548, 732): (6, {z5: p1 + p2, z4: p0 + p3, z3: p4 + p5, z2: p5, z1: 3*p2 + 2*p3 + p4 + p5}, (2*p0 + 3*p1, 3*p2 + 2*p3, p4, p5), (0, p0 + p1 + 4*p2 + 3*p3 + 2*p4 + 2*p5, 0, p0 + p1 + p2 + p3, p1 + p2, p1 + p2, 0, p0 + p1 + p2 + p3, p0 + p1 + p2 + p3, 0, p1 + p2, 0, 0, p1 + p2, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + p2 + p3 + p4 + 2*p5, 0, 0, p1 + p2, p0 + p1 + p2 + p3, p0 + p1 + p2 + p3 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5)), (0, 4, 172, 506, 722): (6, {z5: p2 + p5, z4: p1 + p3, z3: p4, z2: 2*p3 + p4 + 3*p5, z1: p0 + 2*p1 + 3*p2 + 2*p3 + p4 + 3*p5}, (p0, 2*p1 + 3*p2, 2*p3 + 3*p5, p4), (p0 + 3*p1 + 4*p2 + 3*p3 + 2*p4 + 4*p5, 0, p2 + p5, 0, p1 + p2 + 3*p3 + 2*p4 + 4*p5, p1 + p2 + p3 + p5, 0, 0, p1 + p2 + p3 + p5, p2 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p3 + p5, 0, 0, 0, 0, p1 + p2 + p3 + p5, p1 + p2 + p3 + p4 + p5, p1 + p2 + p3 + p4 + p5, p2 + p5, p1 + p2 + p3 + p4 + p5, p1 + p2 + p3 + p4 + p5, p2 + p5, p2 + p5, 0, 0)), (0, 1, 29, 238, 716): (6, {z5: p2 + p5, z4: p0 + p1, z3: p0 + 3*p2, z2: p0 + p1 + 3*p2 + p4 + 3*p5, z1: 3*p2 + p3 + p4 + 3*p5}, (p3, p4, p1 + 3*p5, p0 + 3*p2), (4*p2 + p3 + p4 + 4*p5, 2*p0 + 2*p1 + 4*p2 + p4 + 4*p5, p2 + p5, 0, 0, 0, 0, 0, 0, p2 + p5, 0, 0, 0, 0, p0 + p1 + p2 + p5, 0, 0, p0 + p1 + p2 + p5, p2 + p5, 0, 0, p2 + p5, 0, p2 + p5, p2 + p5, p2 + p5, p2 + p5, 0, 0, 2*p0 + p1 + 4*p2 + p5, 0, 0, 0, 0, p0 + p1 + p2 + p5, p0 + p1 + p2 + p5)), (0, 1, 29, 165, 493): (5, {z5: p3, z4: p1 + p2, z3: p1, z2: p4, z1: p0 + p1 + p2 + 2*p3 + p4}, (p0, 2*p3 + p4, p2, p1), (p0 + 2*p1 + 2*p2 + 3*p3 + p4, p4, p1 + p2 + p3, 0, 0, 0, 0, 0, p3, p1 + p2 + p3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p3, 0, 0, 0, 0, p1 + p2 + p3, 0, p3, 2*p1 + p2 + p3, 0, p3, 0, p3, 0, 0)), (0, 4, 172, 487, 722): (6, {z5: p2 + p5, z4: p1 + p3, z3: p4, z2: 2*p3 + p4 + 3*p5, z1: p0 + 2*p1 + 3*p2 + 2*p3 + p4 + 3*p5}, (p0, 2*p1 + 3*p2, 2*p3 + 3*p5, p4), (p0 + 3*p1 + 4*p2 + 3*p3 + 2*p4 + 4*p5, 0, p1 + p2 + p3 + p5, 0, p1 + p2 + 3*p3 + 2*p4 + 4*p5, p2 + p5, 0, 0, p2 + p5, p1 + p2 + p3 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p2 + p5, 0, 0, 0, 0, p2 + p5, p1 + p2 + p3 + p4 + p5, p1 + p2 + p3 + p4 + p5, p2 + p5, p1 + p2 + p3 + p4 + p5, p1 + p2 + p3 + p4 + p5, p1 + p2 + p3 + p5, p1 + p2 + p3 + p5, 0, 0)), (0, 4, 172, 506, 700): (5, {z5: p0, z4: p2 + p3, z3: p4, z2: 2*p3 + p4, z1: p1 + 2*p2 + 2*p3 + p4}, (p1, 2*p2, 2*p3, p4), (p0 + p1 + 3*p2 + 3*p3 + 2*p4, 0, 0, p0, p0 + p2 + 3*p3 + 2*p4, p0 + p2 + p3, 0, 0, p0 + p2 + p3, 0, p0, 0, 0, 0, p0, p0, p0, 0, 0, 0, p0, p0 + p2 + p3, p0, p0, 0, p0, p0 + p2 + p3, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, 0, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, 0, 0, 0, p0)), (0, 1, 4, 172, 506): (6, {z5: p4 + p5, z4: p0, z3: p0 + p3 + 2*p4, z2: p2 + p3, z1: p0 + p1 + p2 + p3 + 2*p4 + 2*p5}, (p1, p2 + 2*p5, p3 + 2*p4, p0), (2*p0 + p1 + p2 + p3 + 3*p4 + 3*p5, p2 + p3, 0, 0, 2*p0 + p3 + 3*p4 + p5, p4 + p5, 0, 0, p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p4 + p5, 0, 0, 0, 0, p4 + p5, p0 + p4 + p5, p0 + p4 + p5, 0, p0 + p4 + p5, p0 + p4 + p5, 0, 0, 0, 0)), (0, 1, 4, 158, 473): (5, {z5: p3, z4: p1, z3: p4, z2: p1 + p2 + 2*p3 + p4, z1: p0 + p1 + p2 + 2*p3 + p4}, (p0, p2, 2*p3 + p4, p1), (p0 + 2*p1 + p2 + 3*p3 + p4, 2*p1 + p2 + 3*p3 + p4, p3, 0, p4, 0, 0, 0, 0, p3, 0, 0, 0, 0, p3, 0, 0, p3, p1 + p3, 0, 0, 0, 0, p1 + p3, p1 + p3, p1 + p3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)), (0, 4, 29, 309, 613): (5, {z5: p3, z4: p0 + p1 + p2, z3: p1, z2: p1 + p2, z1: p4}, (2*p3 + p4, p0, p2, p1), (p4, 0, 0, p3, p0 + 2*p1 + 2*p2 + p3, p0 + p1 + p2 + p3, 0, 0, p0 + p1 + p2 + p3, 0, p3, 0, 0, 0, p3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + 2*p1 + p2 + p3, 0, 0, p0 + p1 + p2 + p3, p0 + p1 + p2 + p3, p3, 0)), (0, 1, 158, 473, 716): (5, {z5: p3, z4: p4, z3: p1, z2: p1 + p2 + 3*p3 + 2*p4, z1: p0 + p1 + p2 + 3*p3 + 2*p4}, (p0, p2, 3*p3 + 2*p4, p1), (p0 + 2*p1 + p2 + 4*p3 + 3*p4, 2*p1 + p2 + 4*p3 + 3*p4, p3 + p4, 0, 0, 0, 0, 0, 0, p3 + p4, 0, 0, 0, 0, p3 + p4, 0, 0, p3 + p4, p1 + p3 + p4, 0, 0, p3, 0, p1 + p3 + p4, p1 + p3 + p4, p1 + p3 + p4, p3, 0, 0, p3, 0, 0, 0, 0, p3, p3)), (0, 1, 158, 482, 716): (5, {z5: p3, z4: p4, z3: p1, z2: p1 + p2 + 3*p3 + 2*p4, z1: p0 + p1 + p2 + 3*p3 + 2*p4}, (p0, p2, 3*p3 + 2*p4, p1), (p0 + 2*p1 + p2 + 4*p3 + 3*p4, 2*p1 + p2 + 4*p3 + 3*p4, p3, 0, 0, 0, 0, 0, 0, p3, 0, 0, 0, 0, p3, 0, 0, p3, p1 + p3 + p4, 0, 0, p3 + p4, 0, p1 + p3 + p4, p1 + p3 + p4, p1 + p3 + p4, p3 + p4, 0, 0, p3, 0, 0, 0, 0, p3 + p4, p3 + p4)), (0, 4, 29, 309, 722): (7, {z5: p3 + p4 + p5, z4: p0 + p1 + p2, z3: p2 + 3*p3, z2: p1 + p2 + 3*p3 + 3*p5, z1: 3*p3 + 3*p4 + 3*p5 + p6}, (p6, p0 + 3*p4, p1 + 3*p5, p2 + 3*p3), (4*p3 + 4*p4 + 4*p5 + p6, 0, p3 + p4 + p5, 0, p0 + 2*p1 + 2*p2 + 4*p3 + p4 + 4*p5, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, p0 + p1 + p2 + p3 + p4 + p5, p3 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p3 + p4 + p5, 0, 0, 0, 0, p3 + p4 + p5, p3 + p4 + p5, p3 + p4 + p5, p0 + p1 + 2*p2 + 4*p3 + p4 + p5, p3 + p4 + p5, p3 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5, 0, 0)), (1, 4, 204, 540, 732): (7, {z5: p0 + p2 + p4, z4: p1 + p5 + p6, z3: p3, z2: p3 + 3*p4 + 2*p5, z1: 3*p2 + p3 + 3*p4 + 2*p5 + 2*p6}, (3*p0 + 2*p1, 3*p2 + 2*p6, 3*p4 + 2*p5, p3), (0, p0 + p1 + 4*p2 + 2*p3 + 4*p4 + 3*p5 + 3*p6, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p1 + p2 + 2*p3 + 4*p4 + 3*p5 + p6, p0 + p1 + p2 + p4 + p5 + p6, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p1 + p2 + p4 + p5 + p6, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, 0, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p1 + p2 + p4 + p5 + p6, 0, 0, p0 + p1 + p2 + p4 + p5 + p6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p2 + p4, 0, 0, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4)), (4, 29, 309, 613, 712): (5, {z5: p1, z4: p0, z3: p2 + p3 + p4, z2: p3, z1: p3 + p4}, (2*p0, p2, p4, p3), (0, 0, 0, p0 + p1, p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p1 + p2 + p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p1, p0 + p1, 0, p1, 0, p0 + p1, p1, 0, 0, 0, p1, p1, p1, 0, p1, p1, 0, 0, p1, p1, p0 + p1 + p2 + 2*p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4, p0 + p1, 0)), (0, 1, 4, 172, 717): (5, {z5: p3, z4: p1, z3: p1 + p2 + 3*p3, z2: p2 + 3*p3 + p4, z1: p0 + p1 + p2 + 3*p3 + p4}, (p0, p4, p2, p1 + 3*p3), (p0 + 2*p1 + p2 + 4*p3 + p4, p2 + 4*p3 + p4, 0, p3, 2*p1 + p2 + 4*p3, 0, 0, p3, 0, 0, p3, 0, 0, p3, 0, 0, 0, 0, p3, 0, 0, 0, 0, p3, p3, p3, 0, p1 + p3, p1 + p3, 0, p1 + p3, p1 + p3, 0, 0, 0, 0)), (1, 29, 238, 563, 709): (5, {z5: p0, z4: p1 + p3, z3: p2 + p4, z2: p2, z1: p2 + 2*p3 + p4}, (2*p1, 2*p3, p4, p2), (0, p0 + p1 + 2*p2 + 3*p3 + 2*p4, 0, 0, 0, p0 + p1 + p3, p0, 0, 0, p0, p0 + p1 + p3, 0, 0, p0 + p1 + p3, p0 + p1 + p2 + p3 + p4, p0, 0, p0 + p1 + p2 + p3 + p4, 0, p0, 0, p0, p0, 0, p0, p0, 0, p0, 0, p0 + p1 + 2*p2 + p3 + p4, 0, p0, p0 + p1 + p3, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4)), (0, 1, 29, 158, 716): (5, {z5: p2 + p3, z4: p4, z3: 3*p3, z2: p1 + 3*p2 + 3*p3 + p4, z1: p0 + p1 + 3*p2 + 3*p3 + p4}, (p0, p1, 3*p2, 3*p3 + p4), (p0 + p1 + 4*p2 + 4*p3 + 2*p4, p1 + 4*p2 + 4*p3 + 2*p4, p2 + p3, 0, 0, 0, 0, 0, 0, p2 + p3, 0, 0, 0, 0, p2 + p3, 0, 0, p2 + p3, p2 + p3 + p4, 0, 0, p2 + p3, 0, p2 + p3 + p4, p2 + p3 + p4, p2 + p3 + p4, p2 + p3, 0, 0, p2 + 4*p3, 0, 0, 0, 0, p2 + p3, p2 + p3)), (0, 1, 29, 238, 563): (6, {z5: p4 + p5, z4: p0 + p1, z3: p1, z2: p0 + p1 + p2 + 2*p5, z1: p2 + p3}, (p3 + 2*p4, p2 + 2*p5, p0, p1), (p2 + p3, 2*p0 + 2*p1 + p2 + p4 + 3*p5, 0, 0, 0, p4 + p5, 0, 0, 0, 0, p4 + p5, 0, 0, p4 + p5, p0 + p1 + p4 + p5, 0, 0, p0 + p1 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + 2*p1 + p4 + p5, 0, 0, p4 + p5, 0, p0 + p1 + p4 + p5, p0 + p1 + p4 + p5)), (0, 1, 4, 158, 717): (5, {z5: p3, z4: p2, z3: 3*p3 + p4, z2: p1 + p2 + 3*p3 + p4, z1: p0 + p1 + p2 + 3*p3 + p4}, (p0, p1, p4, p2 + 3*p3), (p0 + p1 + 2*p2 + 4*p3 + p4, p1 + 2*p2 + 4*p3 + p4, 0, p3, 4*p3 + p4, 0, 0, p3, 0, 0, p3, 0, 0, p3, 0, 0, 0, 0, p2 + p3, 0, 0, 0, 0, p2 + p3, p2 + p3, p2 + p3, 0, p3, p3, 0, p3, p3, 0, 0, 0, 0)), (1, 4, 29, 238, 732): (6, {z5: p0 + p2 + p4 + p5, z4: p1 + p3, z3: p3 + 3*p4, z2: 3*p4 + 3*p5, z1: p1 + 3*p2 + p3 + 3*p4 + 3*p5}, (3*p0, 3*p2, p1 + 3*p5, p3 + 3*p4), (0, p0 + 2*p1 + 4*p2 + 2*p3 + 4*p4 + 4*p5, 0, p0 + p2 + p4 + p5, p0 + p2 + 4*p4 + 4*p5, p0 + p2 + p4 + p5, 0, p0 + p2 + p4 + p5, p0 + p2 + p4 + p5, 0, p0 + p2 + p4 + p5, 0, 0, p0 + p2 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, p0 + p1 + p2 + p3 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + p2 + 2*p3 + 4*p4 + p5, 0, 0, p0 + p2 + p4 + p5, p0 + p2 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5, p0 + p1 + p2 + p3 + p4 + p5)), (0, 4, 29, 165, 722): (6, {z5: p2 + p4 + p5, z4: p1 + p3, z3: p3 + 3*p5, z2: 3*p4 + 3*p5, z1: p0 + p1 + 3*p2 + p3 + 3*p4 + 3*p5}, (p0, 3*p2, p1 + 3*p4, p3 + 3*p5), (p0 + 2*p1 + 4*p2 + 2*p3 + 4*p4 + 4*p5, 0, p1 + p2 + p3 + p4 + p5, 0, p2 + 4*p4 + 4*p5, p2 + p4 + p5, 0, 0, p2 + p4 + p5, p1 + p2 + p3 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p3 + p4 + p5, 0, 0, 0, 0, p1 + p2 + p3 + p4 + p5, p2 + p4 + p5, p2 + p4 + p5, p1 + p2 + 2*p3 + p4 + 4*p5, p2 + p4 + p5, p2 + p4 + p5, p2 + p4 + p5, p2 + p4 + p5, 0, 0)), (0, 1, 4, 172, 692): (5, {z5: p0, z4: p2, z3: p2 + p3, z2: p3 + p4, z1: p1 + p2 + p3 + p4}, (p1, p4, p3, p2), (p0 + p1 + 2*p2 + p3 + p4, p0 + p3 + p4, 0, 0, p0 + 2*p2 + p3, 0, p0, 0, 0, 0, 0, p0, p0, 0, p0, p0, p0, p0, 0, p0, p0, 0, p0, 0, 0, 0, 0, p0 + p2, p0 + p2, p0, p0 + p2, p0 + p2, 0, 0, p0, p0)), (1, 29, 238, 548, 704): (5, {z5: p0, z4: p1 + p3, z3: p2 + p4, z2: p2, z1: p2 + 2*p3 + p4}, (2*p1, 2*p3, p4, p2), (0, p0 + p1 + 2*p2 + 3*p3 + 2*p4, p0, p0 + p1 + p3, 0, 0, p0, p0 + p1 + p3, p0 + p1 + p3, 0, 0, 0, 0, 0, p0 + p1 + p2 + p3 + p4, p0, 0, p0 + p1 + p2 + p3 + p4, p0, p0, 0, 0, p0, p0, 0, 0, p0, p0, 0, p0 + p1 + 2*p2 + p3 + p4, 0, p0, 0, p0 + p1 + p3, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4)), (0, 4, 29, 309, 691): (5, {z5: p1, z4: p2 + p3 + p4, z3: p3, z2: p3 + p4, z1: p0}, (p0, p2, p4, p3), (p0 + p1, p1, 0, 0, p1 + p2 + 2*p3 + 2*p4, p1 + p2 + p3 + p4, p1, 0, p1 + p2 + p3 + p4, 0, 0, p1, p1, 0, 0, p1, p1, 0, p1, p1, p1, 0, p1, p1, p1, p1, 0, 0, 0, p1 + p2 + 2*p3 + p4, 0, 0, p1 + p2 + p3 + p4, p1 + p2 + p3 + p4, 0, 0)), (0, 1, 29, 165, 490): (5, {z5: p3, z4: p1 + p2, z3: p1, z2: p4, z1: p0 + p1 + p2 + 2*p3 + p4}, (p0, 2*p3 + p4, p2, p1), (p0 + 2*p1 + 2*p2 + 3*p3 + p4, p4, p1 + p2 + p3, 0, 0, p3, 0, 0, 0, p1 + p2 + p3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p3, 0, 0, 0, 0, p1 + p2 + p3, p3, 0, 2*p1 + p2 + p3, p3, 0, p3, 0, 0, 0)), (0, 29, 165, 490, 722): (5, {z5: p1, z4: p2, z3: p3 + p4, z2: p3, z1: p0 + 3*p1 + 2*p2 + p3 + p4}, (p0, 3*p1 + 2*p2, p4, p3), (p0 + 4*p1 + 3*p2 + 2*p3 + 2*p4, 0, p1 + p2 + p3 + p4, 0, p1, p1 + p2, 0, 0, p1, p1 + p2 + p3 + p4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p3 + p4, 0, 0, 0, 0, p1 + p2 + p3 + p4, p1 + p2, p1, p1 + p2 + 2*p3 + p4, p1 + p2, p1, p1 + p2, p1, 0, 0)), (1, 4, 204, 543, 703): (5, {z5: p0, z4: p1 + p2 + p4, z3: p3, z2: p3 + 2*p4, z1: 2*p2 + p3 + 2*p4}, (2*p1, 2*p2, 2*p4, p3), (0, p0 + p1 + 3*p2 + 2*p3 + 3*p4, p0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + 2*p3 + 3*p4, 0, p0, p0 + p1 + p2 + p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p0, 0, p0 + p1 + p2 + p3 + p4, 0, p0, 0, 0, 0, 0, p0, p0, 0, 0, p0, p0, 0, 0, 0, 0, p0, p0, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4)), (0, 4, 29, 309, 636): (5, {z5: p3, z4: p0 + p1 + p2, z3: p1, z2: p1 + p2, z1: p4}, (2*p3 + p4, p0, p2, p1), (p4, 0, 0, 0, p0 + 2*p1 + 2*p2 + p3, p0 + p1 + p2 + p3, 0, p3, p0 + p1 + p2 + p3, 0, 0, 0, 0, p3, 0, 0, 0, p3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + 2*p1 + p2 + p3, 0, 0, p0 + p1 + p2 + p3, p0 + p1 + p2 + p3, 0, p3)), (0, 29, 165, 490, 695): (5, {z5: p0, z4: p2, z3: p3 + p4, z2: p3, z1: p1 + 2*p2 + p3 + p4}, (p1, 2*p2, p4, p3), (p0 + p1 + 3*p2 + 2*p3 + 2*p4, 0, p0 + p2 + p3 + p4, p0, 0, p0 + p2, p0, 0, 0, p0 + p2 + p3 + p4, 0, 0, p0, p0, 0, 0, 0, p0, 0, 0, p0, p0 + p2 + p3 + p4, p0, p0, 0, p0, p0 + p2 + p3 + p4, p0 + p2, 0, p0 + p2 + 2*p3 + p4, p0 + p2, 0, p0 + p2, 0, 0, p0)), (4, 29, 309, 636, 713): (5, {z5: p1, z4: p0, z3: p2 + p3 + p4, z2: p3, z1: p3 + p4}, (2*p0, p2, p4, p3), (0, 0, 0, 0, p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p1 + p2 + p3 + p4, p1, p0 + p1, p0 + p1 + p2 + p3 + p4, p1, 0, p1, 0, p0 + p1, 0, 0, p1, p0 + p1, p1, 0, 0, p1, p1, 0, 0, p1, 0, p1, p1, p0 + p1 + p2 + 2*p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4, 0, p0 + p1)), (0, 1, 4, 204, 543): (7, {z5: p4 + p5 + p6, z4: p0, z3: p0 + p3 + 2*p4, z2: p0 + p2 + p3 + 2*p4 + 2*p6, z1: p1 + p2 + p3}, (p1 + 2*p5, p2 + 2*p6, p3 + 2*p4, p0), (p1 + p2 + p3, 2*p0 + p2 + p3 + 3*p4 + p5 + 3*p6, 0, p0 + p4 + p5 + p6, 2*p0 + p3 + 3*p4 + p5 + p6, 0, 0, p0 + p4 + p5 + p6, 0, 0, p0 + p4 + p5 + p6, 0, 0, p0 + p4 + p5 + p6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p4 + p5 + p6, p4 + p5 + p6, p4 + p5 + p6, p4 + p5 + p6)), (0, 1, 4, 158, 482): (5, {z5: p3, z4: p1, z3: p4, z2: p1 + p2 + 2*p3 + p4, z1: p0 + p1 + p2 + 2*p3 + p4}, (p0, p2, 2*p3 + p4, p1), (p0 + 2*p1 + p2 + 3*p3 + p4, 2*p1 + p2 + 3*p3 + p4, 0, 0, p4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p3, 0, 0, p3, 0, p1 + p3, p1 + p3, p1 + p3, p3, 0, 0, 0, 0, 0, 0, 0, p3, p3)), (0, 1, 158, 473, 689): (5, {z5: p0, z4: p2, z3: p4, z2: 2*p2 + p3 + p4, z1: p1 + 2*p2 + p3 + p4}, (p1, p3, 2*p2, p4), (p0 + p1 + 3*p2 + p3 + 2*p4, p0 + 3*p2 + p3 + 2*p4, p0 + p2, 0, 0, p0, 0, 0, 0, p0 + p2, p0, p0, 0, p0, p0 + p2, 0, 0, p0 + p2, p0 + p2 + p4, p0, p0, 0, p0, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, 0, 0, p0, 0, 0, p0, 0, p0, 0, 0)), (4, 29, 309, 613, 710): (5, {z5: p1, z4: p0, z3: p2 + p3 + p4, z2: p3, z1: p3 + p4}, (2*p0, p2, p4, p3), (0, 0, p1, p0 + p1, p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p1 + p2 + p3 + p4, p1, 0, p0 + p1 + p2 + p3 + p4, 0, p0 + p1, p1, 0, 0, p0 + p1, 0, p1, 0, p1, 0, 0, 0, p1, 0, 0, p1, p1, 0, 0, p0 + p1 + p2 + 2*p3 + p4, p1, p1, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4, p0 + p1, 0)), (1, 29, 238, 563, 705): (5, {z5: p0, z4: p1 + p3, z3: p2 + p4, z2: p2, z1: p2 + 2*p3 + p4}, (2*p1, 2*p3, p4, p2), (0, p0 + p1 + 2*p2 + 3*p3 + 2*p4, p0, 0, 0, p0 + p1 + p3, 0, 0, 0, 0, p0 + p1 + p3, p0, p0, p0 + p1 + p3, p0 + p1 + p2 + p3 + p4, 0, p0, p0 + p1 + p2 + p3 + p4, p0, 0, p0, 0, 0, p0, 0, 0, p0, 0, p0, p0 + p1 + 2*p2 + p3 + p4, p0, 0, p0 + p1 + p3, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4)), (0, 29, 165, 493, 722): (5, {z5: p1, z4: p2, z3: p3 + p4, z2: p3, z1: p0 + 3*p1 + 2*p2 + p3 + p4}, (p0, 3*p1 + 2*p2, p4, p3), (p0 + 4*p1 + 3*p2 + 2*p3 + 2*p4, 0, p1 + p2 + p3 + p4, 0, p1, p1, 0, 0, p1 + p2, p1 + p2 + p3 + p4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p3 + p4, 0, 0, 0, 0, p1 + p2 + p3 + p4, p1, p1 + p2, p1 + p2 + 2*p3 + p4, p1, p1 + p2, p1, p1 + p2, 0, 0)), (0, 4, 172, 506, 701): (5, {z5: p0, z4: p2 + p3, z3: p4, z2: 2*p3 + p4, z1: p1 + 2*p2 + 2*p3 + p4}, (p1, 2*p2, 2*p3, p4), (p0 + p1 + 3*p2 + 3*p3 + 2*p4, 0, 0, 0, p0 + p2 + 3*p3 + 2*p4, p0 + p2 + p3, p0, p0, p0 + p2 + p3, 0, 0, p0, p0, p0, 0, 0, 0, p0, p0, p0, 0, p0 + p2 + p3, 0, 0, p0, 0, p0 + p2 + p3, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, 0, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, 0, 0, p0, 0)), (0, 4, 172, 487, 694): (5, {z5: p0, z4: p2 + p3, z3: p4, z2: 2*p3 + p4, z1: p1 + 2*p2 + 2*p3 + p4}, (p1, 2*p2, 2*p3, p4), (p0 + p1 + 3*p2 + 3*p3 + 2*p4, 0, p0 + p2 + p3, p0, p0 + p2 + 3*p3 + 2*p4, 0, p0, 0, 0, p0 + p2 + p3, p0, p0, p0, 0, 0, 0, 0, p0, 0, p0, 0, 0, 0, p0, 0, p0, 0, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, 0, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, p0 + p2 + p3, p0 + p2 + p3, p0, 0)), (1, 29, 238, 548, 708): (5, {z5: p0, z4: p1 + p3, z3: p2 + p4, z2: p2, z1: p2 + 2*p3 + p4}, (2*p1, 2*p3, p4, p2), (0, p0 + p1 + 2*p2 + 3*p3 + 2*p4, 0, p0 + p1 + p3, 0, 0, 0, p0 + p1 + p3, p0 + p1 + p3, p0, 0, p0, p0, 0, p0 + p1 + p2 + p3 + p4, 0, p0, p0 + p1 + p2 + p3 + p4, 0, 0, p0, p0, 0, 0, p0, p0, 0, 0, p0, p0 + p1 + 2*p2 + p3 + p4, p0, 0, 0, p0 + p1 + p3, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4)), (0, 4, 29, 172, 722): (5, {z5: p1 + p2 + p4, z4: p3, z3: 3*p4, z2: 3*p2 + p3 + 3*p4, z1: p0 + 3*p1 + 3*p2 + p3 + 3*p4}, (p0, 3*p1, 3*p2, p3 + 3*p4), (p0 + 4*p1 + 4*p2 + 2*p3 + 4*p4, 0, p1 + p2 + p4, 0, p1 + 4*p2 + 2*p3 + 4*p4, p1 + p2 + p4, 0, 0, p1 + p2 + p4, p1 + p2 + p4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p1 + p2 + p4, 0, 0, 0, 0, p1 + p2 + p4, p1 + p2 + p3 + p4, p1 + p2 + p3 + p4, p1 + p2 + 4*p4, p1 + p2 + p3 + p4, p1 + p2 + p3 + p4, p1 + p2 + p4, p1 + p2 + p4, 0, 0)), (1, 4, 204, 543, 732): (7, {z5: p0 + p2 + p4, z4: p1 + p5 + p6, z3: p3, z2: p3 + 3*p4 + 2*p5, z1: 3*p2 + p3 + 3*p4 + 2*p5 + 2*p6}, (3*p0 + 2*p1, 3*p2 + 2*p6, 3*p4 + 2*p5, p3), (0, p0 + p1 + 4*p2 + 2*p3 + 4*p4 + 3*p5 + 3*p6, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p1 + p2 + 2*p3 + 4*p4 + 3*p5 + p6, p0 + p2 + p4, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p2 + p4, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, 0, 0, p0 + p1 + p2 + p3 + p4 + p5 + p6, p0 + p2 + p4, 0, 0, p0 + p2 + p4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p2 + p4, 0, 0, p0 + p1 + p2 + p4 + p5 + p6, p0 + p1 + p2 + p4 + p5 + p6, p0 + p1 + p2 + p4 + p5 + p6, p0 + p1 + p2 + p4 + p5 + p6)), (0, 1, 29, 165, 687): (5, {z5: p0, z4: p3 + p4, z3: p3, z2: p2, z1: p1 + p2 + p3 + p4}, (p1, p2, p4, p3), (p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p2, p0 + p3 + p4, p0, p0, 0, p0, p0, 0, p0 + p3 + p4, p0, p0, p0, p0, 0, p0, p0, 0, 0, p0, p0, p0 + p3 + p4, p0, 0, 0, 0, p0 + p3 + p4, 0, 0, p0 + 2*p3 + p4, 0, 0, 0, 0, 0, 0)), (1, 4, 204, 543, 707): (5, {z5: p0, z4: p1 + p2 + p4, z3: p3, z2: p3 + 2*p4, z1: 2*p2 + p3 + 2*p4}, (2*p1, 2*p2, 2*p4, p3), (0, p0 + p1 + 3*p2 + 2*p3 + 3*p4, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + 2*p3 + 3*p4, 0, 0, p0 + p1 + p2 + p3 + p4, 0, p0, p0 + p1 + p2 + p3 + p4, 0, p0, p0 + p1 + p2 + p3 + p4, 0, 0, p0, 0, p0, p0, 0, 0, p0, p0, 0, 0, p0, p0, p0, 0, 0, 0, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4)), (0, 1, 158, 482, 690): (5, {z5: p0, z4: p2, z3: p4, z2: 2*p2 + p3 + p4, z1: p1 + 2*p2 + p3 + p4}, (p1, p3, 2*p2, p4), (p0 + p1 + 3*p2 + p3 + 2*p4, p0 + 3*p2 + p3 + 2*p4, 0, p0, 0, p0, 0, p0, 0, 0, 0, p0, 0, 0, 0, 0, 0, 0, p0 + p2 + p4, p0, p0, p0 + p2, p0, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, p0 + p2, p0, 0, 0, p0, 0, 0, p0, p0 + p2, p0 + p2)), (0, 1, 4, 172, 487): (6, {z5: p4 + p5, z4: p0, z3: p0 + p3 + 2*p4, z2: p2 + p3, z1: p0 + p1 + p2 + p3 + 2*p4 + 2*p5}, (p1, p2 + 2*p5, p3 + 2*p4, p0), (2*p0 + p1 + p2 + p3 + 3*p4 + 3*p5, p2 + p3, p4 + p5, 0, 2*p0 + p3 + 3*p4 + p5, 0, 0, 0, 0, p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p4 + p5, p0 + p4 + p5, 0, p0 + p4 + p5, p0 + p4 + p5, p4 + p5, p4 + p5, 0, 0)), (0, 1, 4, 158, 691): (5, {z5: p0, z4: p2, z3: p4, z2: p2 + p3 + p4, z1: p1 + p2 + p3 + p4}, (p1, p3, p4, p2), (p0 + p1 + 2*p2 + p3 + p4, p0 + 2*p2 + p3 + p4, 0, 0, p0 + p4, p0, p0, 0, p0, 0, 0, p0, p0, 0, 0, p0, p0, 0, p0 + p2, p0, p0, 0, p0, p0 + p2, p0 + p2, p0 + p2, 0, 0, 0, p0, 0, 0, p0, p0, 0, 0)), (4, 29, 309, 636, 711): (5, {z5: p1, z4: p0, z3: p2 + p3 + p4, z2: p3, z1: p3 + p4}, (2*p0, p2, p4, p3), (0, 0, p1, 0, p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p1 + p2 + p3 + p4, 0, p0 + p1, p0 + p1 + p2 + p3 + p4, 0, 0, 0, p1, p0 + p1, 0, p1, 0, p0 + p1, 0, p1, p1, 0, 0, p1, p1, 0, p1, 0, 0, p0 + p1 + p2 + 2*p3 + p4, p1, p1, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4, 0, p0 + p1)), (0, 1, 29, 238, 692): (5, {z5: p0, z4: p1 + p2, z3: p1, z2: p1 + p2 + p4, z1: p3 + p4}, (p3, p4, p2, p1), (p0 + p3 + p4, p0 + 2*p1 + 2*p2 + p4, 0, 0, p0, 0, p0, 0, 0, 0, 0, p0, p0, 0, p0 + p1 + p2, p0, p0, p0 + p1 + p2, 0, p0, p0, 0, p0, 0, 0, 0, 0, p0, p0, p0 + 2*p1 + p2, p0, p0, 0, 0, p0 + p1 + p2, p0 + p1 + p2)), (0, 29, 165, 493, 696): (5, {z5: p0, z4: p2, z3: p3 + p4, z2: p3, z1: p1 + 2*p2 + p3 + p4}, (p1, 2*p2, p4, p3), (p0 + p1 + 3*p2 + 2*p3 + 2*p4, 0, p0 + p2 + p3 + p4, p0, 0, 0, p0, 0, p0 + p2, p0 + p2 + p3 + p4, 0, 0, p0, p0, p0, 0, 0, 0, p0, 0, p0, p0 + p2 + p3 + p4, p0, 0, p0, 0, p0 + p2 + p3 + p4, 0, p0 + p2, p0 + p2 + 2*p3 + p4, 0, p0 + p2, 0, p0 + p2, p0, 0)), (0, 4, 172, 487, 697): (5, {z5: p0, z4: p2 + p3, z3: p4, z2: 2*p3 + p4, z1: p1 + 2*p2 + 2*p3 + p4}, (p1, 2*p2, 2*p3, p4), (p0 + p1 + 3*p2 + 3*p3 + 2*p4, 0, p0 + p2 + p3, 0, p0 + p2 + 3*p3 + 2*p4, 0, 0, p0, 0, p0 + p2 + p3, 0, 0, 0, p0, p0, p0, p0, 0, p0, 0, p0, 0, p0, 0, p0, 0, 0, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, 0, p0 + p2 + p3 + p4, p0 + p2 + p3 + p4, p0 + p2 + p3, p0 + p2 + p3, 0, p0)), (0, 29, 165, 493, 699): (5, {z5: p0, z4: p2, z3: p3 + p4, z2: p3, z1: p1 + 2*p2 + p3 + p4}, (p1, 2*p2, p4, p3), (p0 + p1 + 3*p2 + 2*p3 + 2*p4, 0, p0 + p2 + p3 + p4, 0, 0, 0, 0, p0, p0 + p2, p0 + p2 + p3 + p4, p0, p0, 0, 0, 0, p0, p0, p0, 0, p0, 0, p0 + p2 + p3 + p4, 0, p0, 0, p0, p0 + p2 + p3 + p4, 0, p0 + p2, p0 + p2 + 2*p3 + p4, 0, p0 + p2, 0, p0 + p2, 0, p0)), (0, 1, 4, 204, 540): (7, {z5: p4 + p5 + p6, z4: p0, z3: p0 + p3 + 2*p4, z2: p0 + p2 + p3 + 2*p4 + 2*p6, z1: p1 + p2 + p3}, (p1 + 2*p5, p2 + 2*p6, p3 + 2*p4, p0), (p1 + p2 + p3, 2*p0 + p2 + p3 + 3*p4 + p5 + 3*p6, 0, p0 + p4 + p5 + p6, 2*p0 + p3 + 3*p4 + p5 + p6, p4 + p5 + p6, 0, p0 + p4 + p5 + p6, p4 + p5 + p6, 0, p0 + p4 + p5 + p6, 0, 0, p0 + p4 + p5 + p6, p4 + p5 + p6, 0, 0, p4 + p5 + p6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)), (1, 4, 29, 204, 732): (5, {z5: p0 + p1 + p3 + p4, z4: p2, z3: 3*p3, z2: p2 + 3*p3 + 3*p4, z1: 3*p1 + p2 + 3*p3 + 3*p4}, (3*p0, 3*p1, 3*p4, p2 + 3*p3), (0, p0 + 4*p1 + 2*p2 + 4*p3 + 4*p4, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + 2*p2 + 4*p3 + 4*p4, p0 + p1 + p3 + p4, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p3 + p4, 0, p0 + p1 + p2 + p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p3 + p4, 0, 0, p0 + p1 + p3 + p4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + 4*p3 + p4, 0, 0, p0 + p1 + p3 + p4, p0 + p1 + p3 + p4, p0 + p1 + p3 + p4, p0 + p1 + p3 + p4)), (1, 4, 204, 540, 702): (5, {z5: p0, z4: p1 + p2 + p4, z3: p3, z2: p3 + 2*p4, z1: 2*p2 + p3 + 2*p4}, (2*p1, 2*p2, 2*p4, p3), (0, p0 + p1 + 3*p2 + 2*p3 + 3*p4, p0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + 2*p3 + 3*p4, p0 + p1 + p2 + p4, p0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p4, 0, p0 + p1 + p2 + p3 + p4, p0, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p4, p0, 0, p0 + p1 + p2 + p4, p0, 0, p0, p0, 0, p0, 0, 0, 0, p0, p0, 0, 0, 0, 0, 0, 0, 0)), (0, 1, 158, 482, 693): (5, {z5: p0, z4: p2, z3: p4, z2: 2*p2 + p3 + p4, z1: p1 + 2*p2 + p3 + p4}, (p1, p3, 2*p2, p4), (p0 + p1 + 3*p2 + p3 + 2*p4, p0 + 3*p2 + p3 + 2*p4, 0, 0, 0, 0, p0, 0, p0, 0, p0, 0, p0, p0, 0, p0, p0, 0, p0 + p2 + p4, 0, 0, p0 + p2, 0, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, p0 + p2, 0, p0, 0, 0, p0, p0, 0, p0 + p2, p0 + p2)), (0, 1, 29, 238, 548): (6, {z5: p4 + p5, z4: p0 + p1, z3: p1, z2: p0 + p1 + p2 + 2*p5, z1: p2 + p3}, (p3 + 2*p4, p2 + 2*p5, p0, p1), (p2 + p3, 2*p0 + 2*p1 + p2 + p4 + 3*p5, 0, p4 + p5, 0, 0, 0, p4 + p5, p4 + p5, 0, 0, 0, 0, 0, p0 + p1 + p4 + p5, 0, 0, p0 + p1 + p4 + p5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + 2*p1 + p4 + p5, 0, 0, 0, p4 + p5, p0 + p1 + p4 + p5, p0 + p1 + p4 + p5)), (0, 29, 165, 490, 698): (5, {z5: p0, z4: p2, z3: p3 + p4, z2: p3, z1: p1 + 2*p2 + p3 + p4}, (p1, 2*p2, p4, p3), (p0 + p1 + 3*p2 + 2*p3 + 2*p4, 0, p0 + p2 + p3 + p4, 0, 0, p0 + p2, 0, p0, 0, p0 + p2 + p3 + p4, p0, p0, 0, 0, p0, p0, p0, 0, p0, p0, 0, p0 + p2 + p3 + p4, 0, 0, p0, 0, p0 + p2 + p3 + p4, p0 + p2, 0, p0 + p2 + 2*p3 + p4, p0 + p2, 0, p0 + p2, 0, p0, 0)), (0, 1, 29, 165, 716): (6, {z5: p4 + p5, z4: p1 + p2, z3: p1 + 3*p4, z2: p3 + 3*p4 + 3*p5, z1: p0 + p1 + p2 + p3 + 3*p4 + 3*p5}, (p0, p3, p2 + 3*p5, p1 + 3*p4), (p0 + 2*p1 + 2*p2 + p3 + 4*p4 + 4*p5, p3 + 4*p4 + 4*p5, p1 + p2 + p4 + p5, 0, 0, 0, 0, 0, 0, p1 + p2 + p4 + p5, 0, 0, 0, 0, p4 + p5, 0, 0, p4 + p5, p4 + p5, 0, 0, p1 + p2 + p4 + p5, 0, p4 + p5, p4 + p5, p4 + p5, p1 + p2 + p4 + p5, 0, 0, 2*p1 + p2 + 4*p4 + p5, 0, 0, 0, 0, p4 + p5, p4 + p5)), (0, 1, 4, 204, 687): (5, {z5: p0, z4: p1, z3: p1 + p3, z2: p1 + p2 + p3, z1: p2 + p3 + p4}, (p4, p2, p3, p1), (p0 + p2 + p3 + p4, p0 + 2*p1 + p2 + p3, p0, p0 + p1, p0 + 2*p1 + p3, 0, p0, p0 + p1, 0, p0, p0 + p1, p0, p0, p0 + p1, 0, p0, p0, 0, 0, p0, p0, p0, p0, 0, 0, 0, p0, 0, 0, p0, 0, 0, 0, 0, 0, 0)), (4, 29, 309, 636, 732): (5, {z5: p0, z4: p1, z3: p2 + p3 + p4, z2: p3, z1: p3 + p4}, (3*p0 + 2*p1, p2, p4, p3), (0, p0, 0, p0, p0 + p1 + p2 + 2*p3 + 2*p4, p0 + p1 + p2 + p3 + p4, 0, p0 + p1, p0 + p1 + p2 + p3 + p4, 0, p0, 0, 0, p0 + p1, p0, 0, 0, p0 + p1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, p0 + p1 + p2 + 2*p3 + p4, 0, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p3 + p4, p0, p0 + p1)), (1, 4, 204, 540, 706): (5, {z5: p0, z4: p1 + p2 + p4, z3: p3, z2: p3 + 2*p4, z1: 2*p2 + p3 + 2*p4}, (2*p1, 2*p2, 2*p4, p3), (0, p0 + p1 + 3*p2 + 2*p3 + 3*p4, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + 2*p3 + 3*p4, p0 + p1 + p2 + p4, 0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p4, p0, p0 + p1 + p2 + p3 + p4, 0, p0, p0 + p1 + p2 + p3 + p4, p0 + p1 + p2 + p4, 0, p0, p0 + p1 + p2 + p4, 0, p0, 0, 0, p0, 0, p0, p0, p0, 0, 0, 0, p0, p0, 0, 0, 0, 0)), (0, 1, 158, 473, 688): (5, {z5: p0, z4: p2, z3: p4, z2: 2*p2 + p3 + p4, z1: p1 + 2*p2 + p3 + p4}, (p1, p3, 2*p2, p4), (p0 + p1 + 3*p2 + p3 + 2*p4, p0 + 3*p2 + p3 + 2*p4, p0 + p2, p0, 0, 0, p0, p0, p0, p0 + p2, 0, 0, p0, 0, p0 + p2, p0, p0, p0 + p2, p0 + p2 + p4, 0, 0, 0, 0, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, 0, p0, 0, 0, p0, 0, p0, 0, 0, 0))}


################################################################
# Analysis by length of cones with given set of relevant roots #
################################################################

# We record the indices of the six distinct roots in each of the two summands in the Cross37 expression -(Yoshida34 + Yoshida8).

sumY34 = [24, 10, 17, 27, 26, 33]

sumY8 = [9, 5, 7, 23, 31, 34]


############
# Length 9 #
############

# print len9
# [(1, 3, 4, 5, 7, 8, 10, 13, 14, 17, 29, 32, 33, 34, 35)]
# allTuplesForRelevantRoots[len9[0]]
# [(4, 29, 309, 613, 732),
#  (1, 29, 238, 563, 732),
#  (1, 4, 29, 309, 732),
#  (1, 29, 238, 548, 732),
#  (1, 4, 204, 540, 732),
#  (1, 4, 204, 543, 732),
#  (1, 4, 29, 204, 732),
#  (1, 4, 29, 238, 732),
#  (4, 29, 309, 636, 732)]



testPolys9 = {allTuplesForRelevantRoots[len9[0]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len9[0]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len9[0]]))}

allPolys9 = set(testPolys9.values())
allTuplesPerPolys9 = {P : [u for u in testPolys9 if testPolys9[u]==P] for P in allPolys9}
allTuplesPerPolys9.values()
[[(1, 29, 238, 563, 732), (1, 29, 238, 548, 732)],
 [(1, 4, 29, 309, 732)],
 [(1, 4, 29, 238, 732)],
 [(4, 29, 309, 613, 732), (4, 29, 309, 636, 732)],
 [(1, 4, 204, 543, 732), (1, 4, 204, 540, 732)],
 [(1, 4, 29, 204, 732)]]


###############################
# Cone (4, 29, 309, 613, 732) #
###############################

u =  allTuplesForRelevantRoots[len9[0]][0]
print u
(4, 29, 309, 613, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p0 + 2*p1, p2, p4, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0,
 3: p0 + p1,
 4: p0 + p1 + p2 + 2*p3 + 2*p4,
 5: p0 + p1 + p2 + p3 + p4,
 7: p0,
 8: p0 + p1 + p2 + p3 + p4,
 10: p0 + p1,
 13: p0,
 14: p0 + p1,
 17: p0,
 29: p0 + p1 + p2 + 2*p3 + p4,
 32: p0 + p1 + p2 + p3 + p4,
 33: p0 + p1 + p2 + p3 + p4,
 34: p0 + p1,
 35: p0}


###############################
# Cone (1, 29, 238, 563, 732) #
###############################

u =  allTuplesForRelevantRoots[len9[0]][1]
print u
(1, 29, 238, 563, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(2*p0 + 3*p1, 3*p2 + 2*p3, p4, p5)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0 + p1 + 4*p2 + 3*p3 + 2*p4 + 2*p5,
 3: p1 + p2,
 4: p1 + p2,
 5: p0 + p1 + p2 + p3,
 7: p1 + p2,
 8: p1 + p2,
 10: p0 + p1 + p2 + p3,
 13: p0 + p1 + p2 + p3,
 14: p0 + p1 + p2 + p3 + p4 + p5,
 17: p0 + p1 + p2 + p3 + p4 + p5,
 29: p0 + p1 + p2 + p3 + p4 + 2*p5,
 32: p0 + p1 + p2 + p3,
 33: p1 + p2,
 34: p0 + p1 + p2 + p3 + p4 + p5,
 35: p0 + p1 + p2 + p3 + p4 + p5}





#############################
# Cone (1, 4, 29, 309, 732) #
#############################

u =  allTuplesForRelevantRoots[len9[0]][2]
print u
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
7
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p1, p0 + 3*p4, p3 + 3*p5, p2 + 3*p6)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p1 + 4*p4 + 4*p5 + 4*p6,
 3: p1 + p4 + p5 + p6,
 4: p0 + p1 + 2*p2 + 2*p3 + p4 + 4*p5 + 4*p6,
 5: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 7: p1 + p4 + p5 + p6,
 8: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 10: p1 + p4 + p5 + p6,
 13: p1 + p4 + p5 + p6,
 14: p1 + p4 + p5 + p6,
 17: p1 + p4 + p5 + p6,
 29: p0 + p1 + 2*p2 + p3 + p4 + p5 + 4*p6,
 32: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 33: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 34: p1 + p4 + p5 + p6,
 35: p1 + p4 + p5 + p6}

###############################
# Cone (1, 29, 238, 548, 732) #
###############################

u =  allTuplesForRelevantRoots[len9[0]][3]
print u
(1, 29, 238, 548, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(2*p0 + 3*p1, 3*p2 + 2*p3, p4, p5)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0 + p1 + 4*p2 + 3*p3 + 2*p4 + 2*p5,
 3: p0 + p1 + p2 + p3,
 4: p1 + p2,
 5: p1 + p2,
 7: p0 + p1 + p2 + p3,
 8: p0 + p1 + p2 + p3,
 10: p1 + p2,
 13: p1 + p2,
 14: p0 + p1 + p2 + p3 + p4 + p5,
 17: p0 + p1 + p2 + p3 + p4 + p5,
 29: p0 + p1 + p2 + p3 + p4 + 2*p5,
 32: p1 + p2,
 33: p0 + p1 + p2 + p3,
 34: p0 + p1 + p2 + p3 + p4 + p5,
 35: p0 + p1 + p2 + p3 + p4 + p5}

v = allTuplesForRelevantRoots[len9[0]][1]
relevantRoots == {k:dictIneqpsRoots_lower_dim[v][3][k] for k in rootValuesaa2a3a4[v].keys()}
False
[k for k in rootValuesaa2a3a4[v].keys() if relevantRoots[k]!= dictIneqpsRoots_lower_dim[v][3][k]]
[32, 3, 5, 7, 8, 10, 13, 33]

##############################
# Cone (1, 4, 204, 540, 732) #
##############################

u =  allTuplesForRelevantRoots[len9[0]][4]
print u
(1, 4, 204, 540, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
7
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p0 + 2*p1, 3*p2 + 2*p6, 3*p4 + 2*p5, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0 + p1 + 4*p2 + 2*p3 + 4*p4 + 3*p5 + 3*p6,
 3: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 4: p0 + p1 + p2 + 2*p3 + 4*p4 + 3*p5 + p6,
 5: p0 + p1 + p2 + p4 + p5 + p6,
 7: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 8: p0 + p1 + p2 + p4 + p5 + p6,
 10: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 13: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 14: p0 + p1 + p2 + p4 + p5 + p6,
 17: p0 + p1 + p2 + p4 + p5 + p6,
 29: p0 + p2 + p4,
 32: p0 + p2 + p4,
 33: p0 + p2 + p4,
 34: p0 + p2 + p4,
 35: p0 + p2 + p4}

##############################
# Cone (1, 4, 204, 543, 732) #
##############################

u =  allTuplesForRelevantRoots[len9[0]][5]
print u
(1, 4, 204, 543, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
7
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p0 + 2*p1, 3*p2 + 2*p6, 3*p4 + 2*p5, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0 + p1 + 4*p2 + 2*p3 + 4*p4 + 3*p5 + 3*p6,
 3: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 4: p0 + p1 + p2 + 2*p3 + 4*p4 + 3*p5 + p6,
 5: p0 + p2 + p4,
 7: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 8: p0 + p2 + p4,
 10: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 13: p0 + p1 + p2 + p3 + p4 + p5 + p6,
 14: p0 + p2 + p4,
 17: p0 + p2 + p4,
 29: p0 + p2 + p4,
 32: p0 + p1 + p2 + p4 + p5 + p6,
 33: p0 + p1 + p2 + p4 + p5 + p6,
 34: p0 + p1 + p2 + p4 + p5 + p6,
 35: p0 + p1 + p2 + p4 + p5 + p6}


#############################
# Cone (1, 4, 29, 204, 732) #
#############################

u =  allTuplesForRelevantRoots[len9[0]][6]
print u
(1, 4, 29, 204, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p0, 3*p1, 3*p4, p2 + 3*p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0 + 4*p1 + 2*p2 + 4*p3 + 4*p4,
 3: p0 + p1 + p2 + p3 + p4,
 4: p0 + p1 + 2*p2 + 4*p3 + 4*p4,
 5: p0 + p1 + p3 + p4,
 7: p0 + p1 + p2 + p3 + p4,
 8: p0 + p1 + p3 + p4,
 10: p0 + p1 + p2 + p3 + p4,
 13: p0 + p1 + p2 + p3 + p4,
 14: p0 + p1 + p3 + p4,
 17: p0 + p1 + p3 + p4,
 29: p0 + p1 + 4*p3 + p4,
 32: p0 + p1 + p3 + p4,
 33: p0 + p1 + p3 + p4,
 34: p0 + p1 + p3 + p4,
 35: p0 + p1 + p3 + p4}

#############################
# Cone (1, 4, 29, 238, 732) #
#############################

u =  allTuplesForRelevantRoots[len9[0]][7]
print u
(1, 4, 29, 238, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p0, 3*p2, p1 + 3*p5, p3 + 3*p4)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0 + 2*p1 + 4*p2 + 2*p3 + 4*p4 + 4*p5,
 3: p0 + p2 + p4 + p5,
 4: p0 + p2 + 4*p4 + 4*p5,
 5: p0 + p2 + p4 + p5,
 7: p0 + p2 + p4 + p5,
 8: p0 + p2 + p4 + p5,
 10: p0 + p2 + p4 + p5,
 13: p0 + p2 + p4 + p5,
 14: p0 + p1 + p2 + p3 + p4 + p5,
 17: p0 + p1 + p2 + p3 + p4 + p5,
 29: p0 + p1 + p2 + 2*p3 + 4*p4 + p5,
 32: p0 + p2 + p4 + p5,
 33: p0 + p2 + p4 + p5,
 34: p0 + p1 + p2 + p3 + p4 + p5,
 35: p0 + p1 + p2 + p3 + p4 + p5}

###############################
# Cone (4, 29, 309, 636, 732) #
###############################

u =  allTuplesForRelevantRoots[len9[0]][8]
print u
(4, 29, 309, 636, 732)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(3*p0 + 2*p1, p2, p4, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{1: p0,
 3: p0,
 4: p0 + p1 + p2 + 2*p3 + 2*p4,
 5: p0 + p1 + p2 + p3 + p4,
 7: p0 + p1,
 8: p0 + p1 + p2 + p3 + p4,
 10: p0,
 13: p0 + p1,
 14: p0,
 17: p0 + p1,
 29: p0 + p1 + p2 + 2*p3 + p4,
 32: p0 + p1 + p2 + p3 + p4,
 33: p0 + p1 + p2 + p3 + p4,
 34: p0,
 35: p0 + p1}



############
# Length 7 #
############

# print len7
# [(0, 2, 4, 5, 8, 9, 21, 26, 27, 28, 29, 30, 31, 32, 33)]
# allTuplesForRelevantRoots[len7[0]]
[(0, 4, 172, 487, 722),
 (0, 4, 29, 309, 722),
 (0, 4, 29, 165, 722),
 (0, 29, 165, 490, 722),
 (0, 29, 165, 493, 722),
 (0, 4, 29, 172, 722),
 (0, 4, 172, 506, 722)]

# len(len7[0])
# 15

testPolys7 = {allTuplesForRelevantRoots[len7[0]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len7[0]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len7[0]]))}

allPolys7 = set(testPolys7.values())
allTuplesPerPolys7 = {P : [u for u in testPolys7 if testPolys7[u]==P] for P in allPolys7}
allTuplesPerPolys7.values()
[[(0, 4, 29, 309, 722)],
 [(0, 4, 172, 506, 722), (0, 4, 172, 487, 722)],
 [(0, 29, 165, 490, 722), (0, 29, 165, 493, 722)],
 [(0, 4, 29, 165, 722)],
 [(0, 4, 29, 172, 722)]]



##############################
# Cone (0, 4, 172, 487, 722) #
##############################

u =  allTuplesForRelevantRoots[len7[0]][0]
print u
(0, 4, 172, 487, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, 2*p1 + 3*p2, 2*p3 + 3*p5, p4)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 3*p1 + 4*p2 + 3*p3 + 2*p4 + 4*p5,
 2: p1 + p2 + p3 + p5,
 4: p1 + p2 + 3*p3 + 2*p4 + 4*p5,
 5: p2 + p5,
 8: p2 + p5,
 9: p1 + p2 + p3 + p5,
 21: p2 + p5,
 26: p2 + p5,
 27: p1 + p2 + p3 + p4 + p5,
 28: p1 + p2 + p3 + p4 + p5,
 29: p2 + p5,
 30: p1 + p2 + p3 + p4 + p5,
 31: p1 + p2 + p3 + p4 + p5,
 32: p1 + p2 + p3 + p5,
 33: p1 + p2 + p3 + p5}



#############################
# Cone (0, 4, 29, 309, 722) #
#############################

u =  allTuplesForRelevantRoots[len7[0]][1]
print u
(0, 4, 29, 309, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
7
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p6, p0 + 3*p4, p1 + 3*p5, p2 + 3*p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: 4*p3 + 4*p4 + 4*p5 + p6,
 2: p3 + p4 + p5,
 4: p0 + 2*p1 + 2*p2 + 4*p3 + p4 + 4*p5,
 5: p0 + p1 + p2 + p3 + p4 + p5,
 8: p0 + p1 + p2 + p3 + p4 + p5,
 9: p3 + p4 + p5,
 21: p3 + p4 + p5,
 26: p3 + p4 + p5,
 27: p3 + p4 + p5,
 28: p3 + p4 + p5,
 29: p0 + p1 + 2*p2 + 4*p3 + p4 + p5,
 30: p3 + p4 + p5,
 31: p3 + p4 + p5,
 32: p0 + p1 + p2 + p3 + p4 + p5,
 33: p0 + p1 + p2 + p3 + p4 + p5}



#############################
# Cone (0, 4, 29, 165, 722) #
#############################

u =  allTuplesForRelevantRoots[len7[0]][2]
print u
(0, 4, 29, 165, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, 3*p2, p1 + 3*p4, p3 + 3*p5)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 2*p1 + 4*p2 + 2*p3 + 4*p4 + 4*p5,
 2: p1 + p2 + p3 + p4 + p5,
 4: p2 + 4*p4 + 4*p5,
 5: p2 + p4 + p5,
 8: p2 + p4 + p5,
 9: p1 + p2 + p3 + p4 + p5,
 21: p1 + p2 + p3 + p4 + p5,
 26: p1 + p2 + p3 + p4 + p5,
 27: p2 + p4 + p5,
 28: p2 + p4 + p5,
 29: p1 + p2 + 2*p3 + p4 + 4*p5,
 30: p2 + p4 + p5,
 31: p2 + p4 + p5,
 32: p2 + p4 + p5,
 33: p2 + p4 + p5}



###############################
# Cone (0, 29, 165, 490, 722) #
###############################

u =  allTuplesForRelevantRoots[len7[0]][3]
print u
(0, 29, 165, 490, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, 3*p1 + 2*p2, p4, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 4*p1 + 3*p2 + 2*p3 + 2*p4,
 2: p1 + p2 + p3 + p4,
 4: p1,
 5: p1 + p2,
 8: p1,
 9: p1 + p2 + p3 + p4,
 21: p1 + p2 + p3 + p4,
 26: p1 + p2 + p3 + p4,
 27: p1 + p2,
 28: p1,
 29: p1 + p2 + 2*p3 + p4,
 30: p1 + p2,
 31: p1,
 32: p1 + p2,
 33: p1}


###############################
# Cone (0, 29, 165, 493, 722) #
###############################

u =  allTuplesForRelevantRoots[len7[0]][4]
print u
(0, 29, 165, 493, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, 3*p1 + 2*p2, p4, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 4*p1 + 3*p2 + 2*p3 + 2*p4,
 2: p1 + p2 + p3 + p4,
 4: p1,
 5: p1,
 8: p1 + p2,
 9: p1 + p2 + p3 + p4,
 21: p1 + p2 + p3 + p4,
 26: p1 + p2 + p3 + p4,
 27: p1,
 28: p1 + p2,
 29: p1 + p2 + 2*p3 + p4,
 30: p1,
 31: p1 + p2,
 32: p1,
 33: p1 + p2}




#############################
# Cone (0, 4, 29, 172, 722) #
#############################

u =  allTuplesForRelevantRoots[len7[0]][5]
print u
(0, 4, 29, 172, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, 3*p1, 3*p2, p3 + 3*p4)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 4*p1 + 4*p2 + 2*p3 + 4*p4,
 2: p1 + p2 + p4,
 4: p1 + 4*p2 + 2*p3 + 4*p4,
 5: p1 + p2 + p4,
 8: p1 + p2 + p4,
 9: p1 + p2 + p4,
 21: p1 + p2 + p4,
 26: p1 + p2 + p4,
 27: p1 + p2 + p3 + p4,
 28: p1 + p2 + p3 + p4,
 29: p1 + p2 + 4*p4,
 30: p1 + p2 + p3 + p4,
 31: p1 + p2 + p3 + p4,
 32: p1 + p2 + p4,
 33: p1 + p2 + p4}





##############################
# Cone (0, 4, 172, 506, 722) #
##############################

u =  allTuplesForRelevantRoots[len7[0]][6]
print u
(0, 4, 172, 506, 722)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, 2*p1 + 3*p2, 2*p3 + 3*p5, p4)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 3*p1 + 4*p2 + 3*p3 + 2*p4 + 4*p5,
 2: p2 + p5,
 4: p1 + p2 + 3*p3 + 2*p4 + 4*p5,
 5: p1 + p2 + p3 + p5,
 8: p1 + p2 + p3 + p5,
 9: p2 + p5,
 21: p1 + p2 + p3 + p5,
 26: p1 + p2 + p3 + p5,
 27: p1 + p2 + p3 + p4 + p5,
 28: p1 + p2 + p3 + p4 + p5,
 29: p2 + p5,
 30: p1 + p2 + p3 + p4 + p5,
 31: p1 + p2 + p3 + p4 + p5,
 32: p2 + p5,
 33: p2 + p5}


############
# Length 5 #
############

# print len5
# [(0, 1, 2, 9, 14, 17, 18, 21, 23, 24, 25, 26, 29, 34, 35)]
# allTuplesForRelevantRoots[len5[0]]
# [(0, 1, 29, 238, 716),
#  (0, 1, 158, 473, 716),
#  (0, 1, 158, 482, 716),
#  (0, 1, 29, 158, 716),
#  (0, 1, 29, 165, 716)]


# len(len5[0])
# 15


testPolys5 = {allTuplesForRelevantRoots[len5[0]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len5[0]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len5[0]]))}

allPolys5 = set(testPolys5.values())
allTuplesPerPolys5 = {P : [u for u in testPolys5 if testPolys5[u]==P] for P in allPolys5}
allTuplesPerPolys5.values()
[[(0, 1, 29, 165, 716)],
 [(0, 1, 29, 158, 716)],
 [(0, 1, 158, 473, 716), (0, 1, 158, 482, 716)],
 [(0, 1, 29, 238, 716)]]


#############################
# Cone (0, 1, 29, 238, 716) #
#############################

u =  allTuplesForRelevantRoots[len5[0]][0]
print u
(0, 1, 29, 238, 716)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p3, p4, p1 + 3*p5, p0 + 3*p2)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: 4*p2 + p3 + p4 + 4*p5,
 1: 2*p0 + 2*p1 + 4*p2 + p4 + 4*p5,
 2: p2 + p5,
 9: p2 + p5,
 14: p0 + p1 + p2 + p5,
 17: p0 + p1 + p2 + p5,
 18: p2 + p5,
 21: p2 + p5,
 23: p2 + p5,
 24: p2 + p5,
 25: p2 + p5,
 26: p2 + p5,
 29: 2*p0 + p1 + 4*p2 + p5,
 34: p0 + p1 + p2 + p5,
 35: p0 + p1 + p2 + p5}



##############################
# Cone (0, 1, 158, 473, 716) #
##############################

u =  allTuplesForRelevantRoots[len5[0]][1]
print u
(0, 1, 158, 473, 716)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p2, 3*p3 + 2*p4, p1)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 2*p1 + p2 + 4*p3 + 3*p4,
 1: 2*p1 + p2 + 4*p3 + 3*p4,
 2: p3 + p4,
 9: p3 + p4,
 14: p3 + p4,
 17: p3 + p4,
 18: p1 + p3 + p4,
 21: p3,
 23: p1 + p3 + p4,
 24: p1 + p3 + p4,
 25: p1 + p3 + p4,
 26: p3,
 29: p3,
 34: p3,
 35: p3}




##############################
# Cone (0, 1, 158, 482, 716) #
##############################

u =  allTuplesForRelevantRoots[len5[0]][2]
print u
(0, 1, 158, 482, 716)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p2, 3*p3 + 2*p4, p1)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 2*p1 + p2 + 4*p3 + 3*p4,
 1: 2*p1 + p2 + 4*p3 + 3*p4,
 2: p3,
 9: p3,
 14: p3,
 17: p3,
 18: p1 + p3 + p4,
 21: p3 + p4,
 23: p1 + p3 + p4,
 24: p1 + p3 + p4,
 25: p1 + p3 + p4,
 26: p3 + p4,
 29: p3,
 34: p3 + p4,
 35: p3 + p4}




#############################
# Cone (0, 1, 29, 158, 716) #
#############################

u =  allTuplesForRelevantRoots[len5[0]][3]
print u
(0, 1, 29, 158, 716)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p1, 3*p2, 3*p3 + p4)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p1 + 4*p2 + 4*p3 + 2*p4,
 1: p1 + 4*p2 + 4*p3 + 2*p4,
 2: p2 + p3,
 9: p2 + p3,
 14: p2 + p3,
 17: p2 + p3,
 18: p2 + p3 + p4,
 21: p2 + p3,
 23: p2 + p3 + p4,
 24: p2 + p3 + p4,
 25: p2 + p3 + p4,
 26: p2 + p3,
 29: p2 + 4*p3,
 34: p2 + p3,
 35: p2 + p3}


#############################
# Cone (0, 1, 29, 165, 716) #
#############################

u =  allTuplesForRelevantRoots[len5[0]][4]
print u
(0, 1, 29, 165, 716)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
6
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p3, p2 + 3*p5, p1 + 3*p4)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 2*p1 + 2*p2 + p3 + 4*p4 + 4*p5,
 1: p3 + 4*p4 + 4*p5,
 2: p1 + p2 + p4 + p5,
 9: p1 + p2 + p4 + p5,
 14: p4 + p5,
 17: p4 + p5,
 18: p4 + p5,
 21: p1 + p2 + p4 + p5,
 23: p4 + p5,
 24: p4 + p5,
 25: p4 + p5,
 26: p1 + p2 + p4 + p5,
 29: 2*p1 + p2 + 4*p4 + p5,
 34: p4 + p5,
 35: p4 + p5}



############
# Length 3 #
############

# print len3
# [(0, 1, 3, 4, 7, 10, 13, 18, 23, 24, 25, 27, 28, 30, 31)]
# allTuplesForRelevantRoots[len3[0]]
# [(0, 1, 4, 204, 717), (0, 1, 4, 172, 717), (0, 1, 4, 158, 717)]

# len(len3[0])
# 15

testPolys3 = {allTuplesForRelevantRoots[len3[0]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len3[0]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len3[0]]))}

allPolys3 = set(testPolys3.values())
allTuplesPerPolys3 = {P : [u for u in testPolys3 if testPolys3[u]==P] for P in allPolys3}
allTuplesPerPolys3.values()
[[(0, 1, 4, 204, 717)], [(0, 1, 4, 158, 717)], [(0, 1, 4, 172, 717)]]



############################
# Cone (0, 1, 4, 204, 717) #
############################

u =  allTuplesForRelevantRoots[len3[0]][0]
print u
(0, 1, 4, 204, 717)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p4, p1, p2, p0 + 3*p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p1 + p2 + 4*p3 + p4,
 1: 2*p0 + p1 + p2 + 4*p3,
 3: p0 + p3,
 4: 2*p0 + p2 + 4*p3,
 7: p0 + p3,
 10: p0 + p3,
 13: p0 + p3,
 18: p3,
 23: p3,
 24: p3,
 25: p3,
 27: p3,
 28: p3,
 30: p3,
 31: p3}

############################
# Cone (0, 1, 4, 172, 717) #
############################

u =  allTuplesForRelevantRoots[len3[0]][1]
print u
(0, 1, 4, 172, 717)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p4, p2, p1 + 3*p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + 2*p1 + p2 + 4*p3 + p4,
 1: p2 + 4*p3 + p4,
 3: p3,
 4: 2*p1 + p2 + 4*p3,
 7: p3,
 10: p3,
 13: p3,
 18: p3,
 23: p3,
 24: p3,
 25: p3,
 27: p1 + p3,
 28: p1 + p3,
 30: p1 + p3,
 31: p1 + p3}


############################
# Cone (0, 1, 4, 158, 717) #
############################

u =  allTuplesForRelevantRoots[len3[0]][2]
print u
(0, 1, 4, 158, 717)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p1, p4, p2 + 3*p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p1 + 2*p2 + 4*p3 + p4,
 1: p1 + 2*p2 + 4*p3 + p4,
 3: p3,
 4: 4*p3 + p4,
 7: p3,
 10: p3,
 13: p3,
 18: p2 + p3,
 23: p2 + p3,
 24: p2 + p3,
 25: p2 + p3,
 27: p3,
 28: p3,
 30: p3,
 31: p3}


############
# Length 2 #
############

# print len2
# [(0, 1, 4, 5, 6, 8, 11, 12, 15, 16, 18, 19, 20, 22, 23, 24, 25, 29, 32, 33), (0, 1, 4, 6, 11, 12, 14, 15, 16, 17, 19, 20, 22, 27, 28, 29, 30, 31, 34, 35), (0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 19, 20, 21, 22, 26, 29)]
# print len(len2)
# 3
# allTuplesForRelevantRoots[len2[0]]
# [(0, 1, 4, 158, 691), (0, 4, 29, 309, 691)]
# allTuplesForRelevantRoots[len2[1]]
# [(0, 1, 4, 172, 692), (0, 1, 29, 238, 692)]
# allTuplesForRelevantRoots[len2[2]]
# [(0, 1, 4, 204, 687), (0, 1, 29, 165, 687)]

# len(len2[0])
# 20
# len(len2[1])
# 20
# len(len2[2])
# 20

# We divide the calculation into each of the three pairs.

testPolys2_0 = {allTuplesForRelevantRoots[len2[0]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len2[0]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len2[0]]))}

testPolys2_1 = {allTuplesForRelevantRoots[len2[1]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len2[1]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len2[1]]))}

testPolys2_2 = {allTuplesForRelevantRoots[len2[2]][k]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len2[2]][k]]) for k in range(0,len(allTuplesForRelevantRoots[len2[2]]))}

allPolys2 = set(testPolys2_0.values()+testPolys2_1.values()+testPolys2_2.values())
allTuplesPerPolys2 = {P : [u for u in testPolys2_0 if testPolys2_0[u]==P]+[u for u in testPolys2_1 if testPolys2_1[u]==P]+[u for u in testPolys2_2 if testPolys2_2[u]==P] for P in allPolys2}

allTuplesPerPolys2.values()
[[(0, 1, 29, 165, 687)],
 [(0, 4, 29, 309, 691)],
 [(0, 1, 4, 158, 691)],
 [(0, 1, 29, 238, 692)],
 [(0, 1, 4, 172, 692)],
 [(0, 1, 4, 204, 687)]]



##########
# PAIR 0 #
##########

len2[0]
(0, 1, 4, 5, 6, 8, 11, 12, 15, 16, 18, 19, 20, 22, 23, 24, 25, 29, 32, 33)

############################
# Cone (0, 1, 4, 158, 691) #
############################

u =  allTuplesForRelevantRoots[len2[0]][0]
print u
(0, 1, 4, 158, 691)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p1, p3, p4, p2)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p1 + 2*p2 + p3 + p4,
 1: p0 + 2*p2 + p3 + p4,
 4: p0 + p4,
 5: p0,
 6: p0,
 8: p0,
 11: p0,
 12: p0,
 15: p0,
 16: p0,
 18: p0 + p2,
 19: p0,
 20: p0,
 22: p0,
 23: p0 + p2,
 24: p0 + p2,
 25: p0 + p2,
 29: p0,
 32: p0,
 33: p0}

#############################
# Cone (0, 4, 29, 309, 691) #
#############################

u =  allTuplesForRelevantRoots[len2[0]][1]
print u
(0, 4, 29, 309, 691)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p0, p2, p4, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p1,
 1: p1,
 4: p1 + p2 + 2*p3 + 2*p4,
 5: p1 + p2 + p3 + p4,
 6: p1,
 8: p1 + p2 + p3 + p4,
 11: p1,
 12: p1,
 15: p1,
 16: p1,
 18: p1,
 19: p1,
 20: p1,
 22: p1,
 23: p1,
 24: p1,
 25: p1,
 29: p1 + p2 + 2*p3 + p4,
 32: p1 + p2 + p3 + p4,
 33: p1 + p2 + p3 + p4}


##########
# PAIR 1 #
##########

# len2[1]
# (0, 1, 4, 6, 11, 12, 14, 15, 16, 17, 19, 20, 22, 27, 28, 29, 30, 31, 34, 35)

############################
# Cone (0, 1, 4, 172, 692) #
############################

u =  allTuplesForRelevantRoots[len2[1]][0]
print u
(0, 1, 4, 172, 692)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p1, p4, p3, p2)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p1 + 2*p2 + p3 + p4,
 1: p0 + p3 + p4,
 4: p0 + 2*p2 + p3,
 6: p0,
 11: p0,
 12: p0,
 14: p0,
 15: p0,
 16: p0,
 17: p0,
 19: p0,
 20: p0,
 22: p0,
 27: p0 + p2,
 28: p0 + p2,
 29: p0,
 30: p0 + p2,
 31: p0 + p2,
 34: p0,
 35: p0}

#############################
# Cone (0, 1, 29, 238, 692) #
#############################

u =  allTuplesForRelevantRoots[len2[1]][1]
print u
(0, 1, 29, 238, 692)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p3, p4, p2, p1)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p3 + p4,
 1: p0 + 2*p1 + 2*p2 + p4,
 4: p0,
 6: p0,
 11: p0,
 12: p0,
 14: p0 + p1 + p2,
 15: p0,
 16: p0,
 17: p0 + p1 + p2,
 19: p0,
 20: p0,
 22: p0,
 27: p0,
 28: p0,
 29: p0 + 2*p1 + p2,
 30: p0,
 31: p0,
 34: p0 + p1 + p2,
 35: p0 + p1 + p2}



##########
# PAIR 2 #
##########

# len2[2]
# (0, 1, 2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16, 19, 20, 21, 22, 26, 29)

############################
# Cone (0, 1, 4, 204, 687) #
############################

u =  allTuplesForRelevantRoots[len2[2]][0]
print u
(0, 1, 4, 204, 687)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p4, p2, p3, p1)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p2 + p3 + p4,
 1: p0 + 2*p1 + p2 + p3,
 2: p0,
 3: p0 + p1,
 4: p0 + 2*p1 + p3,
 6: p0,
 7: p0 + p1,
 9: p0,
 10: p0 + p1,
 11: p0,
 12: p0,
 13: p0 + p1,
 15: p0,
 16: p0,
 19: p0,
 20: p0,
 21: p0,
 22: p0,
 26: p0,
 29: p0}


#############################
# Cone (0, 1, 29, 165, 687) #
#############################

u =  allTuplesForRelevantRoots[len2[2]][1]
print u
(0, 1, 29, 165, 687)
# Number of rays
dictIneqpsRoots_lower_dim[u][0]
5
# The inequalities in ps
dictIneqpsRoots_lower_dim[u][2]
(p1, p2, p4, p3)
# The coefficients of the relevant roots:
relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
relevantRoots
{0: p0 + p1 + p2 + 2*p3 + 2*p4,
 1: p0 + p2,
 2: p0 + p3 + p4,
 3: p0,
 4: p0,
 6: p0,
 7: p0,
 9: p0 + p3 + p4,
 10: p0,
 11: p0,
 12: p0,
 13: p0,
 15: p0,
 16: p0,
 19: p0,
 20: p0,
 21: p0 + p3 + p4,
 22: p0,
 26: p0 + p3 + p4,
 29: p0 + 2*p3 + p4}




############
# Length 1 #
############

# There are two lengths of relevant roots for the cones in len1.

print set([len(len1[k]) for k in range(0,36)])
set[11,20]

len1_11 = [x for x in len1 if len(x)==11]
len1_20 = [x for x in len1 if len(x)==20]


# Rather than ruling out a cancellation for each of the 36 cones in len1, we identify those cones whose scalars zs lie in the same polyhedron. It turns out that several cones share these polyhedrons. The ones with 11 relevant roots are grouped in pairs, whereas the ones with 24 roots are grouped in quadruples. This reduces our computation to 12 cases. If we use common roots for each of the cones in the same class to identify leading terms of roots in Y34 with those in Y8 with identified valuations, the result will be valid on the whole class. In what follows, we collect the data that is used to prove the identification of leading terms of roots by explicit computations.

#########################
# Length 1 and 11 roots #
#########################

print len(len1_11)
12

testPolys_11 = {allTuplesForRelevantRoots[len1_11[k]][0]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len1_11[k]][0]]) for k in range(0,len(len1_11))}

allPolys1_11 = set(testPolys_11.values())
allTuplesPerPolys1_11 = {P : [u for u in testPolys_11 if testPolys_11[u]==P] for P in allPolys1_11}
allTuplesPerPolys1_11.values()
[[(0, 1, 29, 238, 563), (0, 1, 29, 238, 548)],
 [(0, 1, 4, 204, 540), (0, 1, 4, 204, 543)],
 [(0, 1, 4, 172, 487), (0, 1, 4, 172, 506)],
 [(0, 1, 4, 158, 482), (0, 1, 4, 158, 473)],
 [(0, 1, 29, 165, 490), (0, 1, 29, 165, 493)],
 [(0, 4, 29, 309, 613), (0, 4, 29, 309, 636)]]

# # They come in pairs!
# set([len(x) for x in allTuplesPerPolys1_11.values()])
# {2}

# len(allTuplesPerPolys1_11.keys())
# 6

# We confirm that all pairs have the same inequalities, the same dictionary writing the scalars zs as linear expressions in the parameters ps.
all([all([bool(dictIneqpsRoots_lower_dim[x[0]][k]==dictIneqpsRoots_lower_dim[x[1]][k]) for k in range(0,3)]) for x in allTuplesPerPolys1_11.values()])
True

# Check the roots for each pair are different:
all([False == bool(dictIneqpsRoots_lower_dim[x[0]][3]==dictIneqpsRoots_lower_dim[x[1]][3]) for x in  allTuplesPerPolys1_11.values()])
True

# The next dictionaries encode the similarities and difference between the cones with the same solution polyhedron in z1,...,z5.

rootDifferences1_11 = dict()
rootEqualities1_11 = dict()
paramIneqs1_11 = dict()

for x in allTuplesPerPolys1_11.values():
    print x
    # Write the different roots:
    toCheck = [k for k in range(0,36) if dictIneqpsRoots_lower_dim[x[0]][3][k] != dictIneqpsRoots_lower_dim[x[1]][3][k]]
    rootDifferences1_11[tuple(x)] = {k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in toCheck}
    rootEqualities1_11[tuple(x)] ={k: dictIneqpsRoots_lower_dim[x[0]][3][k] for k in range(0,36) if k not in toCheck if dictIneqpsRoots_lower_dim[x[0]][3][k] !=0}
    paramIneqs1_11[tuple(x)] = [dictIneqpsRoots_lower_dim[x[0]][k] for k in range(0,3)]



allPairsOfTuples1_11 = allTuplesPerPolys1_11.values()



##########
# PAIR 0 #
##########

x = tuple(allPairsOfTuples1_11[0])
print x
((0, 1, 29, 238, 563), (0, 1, 29, 238, 548))

# Number of rays
paramIneqs1_11[x][0]
6
# The inequalities in ps
paramIneqs1_11[x][2]
(p3 + 2*p4, p2 + 2*p5, p0, p1)

# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY34}
{10: (p4 + p5, 0),
 17: (p0 + p1 + p4 + p5, p0 + p1 + p4 + p5),
 24: (0, 0),
 26: (0, 0),
 27: (0, 0),
 33: (0, p4 + p5)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY8}
{5: (p4 + p5, 0),
 7: (0, p4 + p5),
 9: (0, 0),
 23: (0, 0),
 31: (0, 0),
 34: (p0 + p1 + p4 + p5, p0 + p1 + p4 + p5)}

# The common non-zero remaining roots:
{k: rootEqualities1_11[x][k] for k in rootEqualities1_11[x].keys() if k not in sumY34 + sumY8}
{0: p2 + p3,
 1: 2*p0 + 2*p1 + p2 + p4 + 3*p5,
 14: p0 + p1 + p4 + p5,
 17: p0 + p1 + p4 + p5,
 29: p0 + 2*p1 + p4 + p5,
 34: p0 + p1 + p4 + p5,
 35: p0 + p1 + p4 + p5}

# The remaining different roots:
{k: rootDifferences1_11[x][k] for k in rootDifferences1_11[x].keys() if k not in sumY34 + sumY8}
{3: (0, p4 + p5), 8: (0, p4 + p5), 13: (p4 + p5, 0), 32: (p4 + p5, 0)}



#############################
# Cone (0, 1, 29, 238, 563) #
#############################

u =  allPairsOfTuples1_11[0][0]
print u
(0, 1, 29, 238, 563)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots

# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p4 + p5, 17: p0 + p1 + p4 + p5, 24: 0, 26: 0, 27: 0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p4 + p5, 7: 0, 9: 0, 23: 0, 31: 0, 34: p0 + p1 + p4 + p5}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][0] for k in rootDifferences1_11[x].keys()}
{3: 0, 5: p4 + p5, 7: 0, 8: 0, 10: p4 + p5, 13: p4 + p5, 32: p4 + p5, 33: 0}



##############################
# Cone (0, 1, 29, 238, 548)] #
##############################

u =  allPairsOfTuples1_11[0][1]
print u
(0, 1, 29, 238, 548)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots

# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0 + p1 + p4 + p5, 24: 0, 26: 0, 27: 0, 33: p4 + p5}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p4 + p5, 9: 0, 23: 0, 31: 0, 34: p0 + p1 + p4 + p5}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][1] for k in rootDifferences1_11[x].keys()}
{3: p4 + p5, 5: 0, 7: p4 + p5, 8: p4 + p5, 10: 0, 13: 0, 32: 0, 33: p4 + p5}


##########
# PAIR 1 #
##########

x = tuple(allPairsOfTuples1_11[1])
print x
((0, 1, 4, 204, 540), (0, 1, 4, 204, 543))

# Number of rays
paramIneqs1_11[x][0]
7
# The inequalities in ps
paramIneqs1_11[x][2]
(p1 + 2*p5, p2 + 2*p6, p3 + 2*p4, p0)

# Sum Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY34}
{10: (p0 + p4 + p5 + p6, p0 + p4 + p5 + p6),
 17: (p4 + p5 + p6, 0),
 24: (0, 0),
 26: (0, 0),
 27: (0, 0),
 33: (0, p4 + p5 + p6)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY8}
{5: (p4 + p5 + p6, 0),
 7: (p0 + p4 + p5 + p6, p0 + p4 + p5 + p6),
 9: (0, 0),
 23: (0, 0),
 31: (0, 0),
 34: (0, p4 + p5 + p6)}

# The common non-zero remaining roots:
{k: rootEqualities1_11[x][k] for k in rootEqualities1_11[x].keys() if k not in sumY34 + sumY8}
{0: p1 + p2 + p3,
 1: 2*p0 + p2 + p3 + 3*p4 + p5 + 3*p6,
 3: p0 + p4 + p5 + p6,
 4: 2*p0 + p3 + 3*p4 + p5 + p6,
 13: p0 + p4 + p5 + p6}

# The remaining different roots:
{k: rootDifferences1_11[x][k] for k in rootDifferences1_11[x].keys() if k not in sumY34 + sumY8}
{8: (p4 + p5 + p6, 0),
 14: (p4 + p5 + p6, 0),
 32: (0, p4 + p5 + p6),
 35: (0, p4 + p5 + p6)}

############################
# Cone (0, 1, 4, 204, 540) #
############################

u =  allPairsOfTuples1_11[1][0]
print u
(0, 1, 4, 204, 540)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots

# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p4 + p5 + p6, 17: p4 + p5 + p6, 24: 0, 26: 0, 27: 0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p4 + p5 + p6, 7: p0 + p4 + p5 + p6, 9: 0, 23: 0, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][0] for k in rootDifferences1_11[x].keys()}
{5: p4 + p5 + p6,
 8: p4 + p5 + p6,
 14: p4 + p5 + p6,
 17: p4 + p5 + p6,
 32: 0,
 33: 0,
 34: 0,
 35: 0}



############################
# Cone (0, 1, 4, 204, 543) #
############################

u =  allPairsOfTuples1_11[1][1]
print u
(0, 1, 4, 204, 543)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots

# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p4 + p5 + p6, 17: 0, 24: 0, 26: 0, 27: 0, 33: p4 + p5 + p6}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0 + p4 + p5 + p6, 9: 0, 23: 0, 31: 0, 34: p4 + p5 + p6}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][1] for k in rootDifferences1_11[x].keys()}
{5: 0,
 8: 0,
 14: 0,
 17: 0,
 32: p4 + p5 + p6,
 33: p4 + p5 + p6,
 34: p4 + p5 + p6,
 35: p4 + p5 + p6}


##########
# PAIR 2 #
##########

x = tuple(allPairsOfTuples1_11[2])
print x
((0, 1, 4, 172, 487), (0, 1, 4, 172, 506))

# Number of rays
paramIneqs1_11[x][0]
6
# The inequalities in ps
paramIneqs1_11[x][2]
(p1, p2 + 2*p5, p3 + 2*p4, p0)

# Sum Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY34}
{10: (0, 0),
 17: (0, 0),
 24: (0, 0),
 26: (0, p4 + p5),
 27: (p0 + p4 + p5, p0 + p4 + p5),
 33: (p4 + p5, 0)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY8}
{5: (0, p4 + p5),
 7: (0, 0),
 9: (p4 + p5, 0),
 23: (0, 0),
 31: (p0 + p4 + p5, p0 + p4 + p5),
 34: (0, 0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_11[x][k] for k in rootEqualities1_11[x].keys() if k not in sumY34 + sumY8}
{0: 2*p0 + p1 + p2 + p3 + 3*p4 + 3*p5,
 1: p2 + p3,
 4: 2*p0 + p3 + 3*p4 + p5,
 28: p0 + p4 + p5,
 30: p0 + p4 + p5}

# The remaining different roots:
{k: rootDifferences1_11[x][k] for k in rootDifferences1_11[x].keys() if k not in sumY34 + sumY8}
{2: (p4 + p5, 0), 8: (0, p4 + p5), 21: (0, p4 + p5), 32: (p4 + p5, 0)}


############################
# Cone (0, 1, 4, 172, 487) #
############################

u =  allPairsOfTuples1_11[2][0]
print u
(0, 1, 4, 172, 487)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots



# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: 0, 26: 0, 27: p0 + p4 + p5, 33: p4 + p5}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: p4 + p5, 23: 0, 31: p0 + p4 + p5, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][0] for k in rootDifferences1_11[x].keys()}
{2: p4 + p5, 5: 0, 8: 0, 9: p4 + p5, 21: 0, 26: 0, 32: p4 + p5, 33: p4 + p5}


############################
# Cone (0, 1, 4, 172, 506) #
############################

u =  allPairsOfTuples1_11[2][1]
print u
(0, 1, 4, 172, 506)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: 0, 26: p4 + p5, 27: p0 + p4 + p5, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p4 + p5, 7: 0, 9: 0, 23: 0, 31: p0 + p4 + p5, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][1] for k in rootDifferences1_11[x].keys()}
{2: 0, 5: p4 + p5, 8: p4 + p5, 9: 0, 21: p4 + p5, 26: p4 + p5, 32: 0, 33: 0}



##########
# PAIR 3 #
##########

x = tuple(allPairsOfTuples1_11[3])
print x
((0, 1, 4, 158, 482), (0, 1, 4, 158, 473))

# Number of rays
paramIneqs1_11[x][0]
5
# The inequalities in ps
paramIneqs1_11[x][2]
(p0, p2, 2*p3 + p4, p1)

# Sum Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY34}
{10: (0, 0),
 17: (0, p3),
 24: (p1 + p3, p1 + p3),
 26: (p3, 0),
 27: (0, 0),
 33: (0, 0)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY8}
{5: (0, 0),
 7: (0, 0),
 9: (0, p3),
 23: (p1 + p3, p1 + p3),
 31: (0, 0),
 34: (p3, 0)}

# The common non-zero remaining roots:
{k: rootEqualities1_11[x][k] for k in rootEqualities1_11[x].keys() if k not in sumY34 + sumY8}
{0: p0 + 2*p1 + p2 + 3*p3 + p4,
 1: 2*p1 + p2 + 3*p3 + p4,
 4: p4,
 18: p1 + p3,
 25: p1 + p3}

# The remaining different roots:
{k: rootDifferences1_11[x][k] for k in rootDifferences1_11[x].keys() if k not in sumY34 + sumY8}
{2: (0, p3), 14: (0, p3), 21: (p3, 0), 35: (p3, 0)}

############################
# Cone (0, 1, 4, 158, 482) #
############################

u =  allPairsOfTuples1_11[3][0]
print u
(0, 1, 4, 158, 482)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: p1 + p3, 26: p3, 27: 0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: 0, 23: p1 + p3, 31: 0, 34: p3}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][0] for k in rootDifferences1_11[x].keys()}
{2: 0, 9: 0, 14: 0, 17: 0, 21: p3, 26: p3, 34: p3, 35: p3}


############################
# Cone (0, 1, 4, 158, 473) #
############################

u =  allPairsOfTuples1_11[3][1]
print u
(0, 1, 4, 158, 473)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p3, 24: p1 + p3, 26: 0, 27: 0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: p3, 23: p1 + p3, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][1] for k in rootDifferences1_11[x].keys()}
{2: p3, 9: p3, 14: p3, 17: p3, 21: 0, 26: 0, 34: 0, 35: 0}


##########
# PAIR 4 #
##########

x = tuple(allPairsOfTuples1_11[4])
print x
((0, 1, 29, 165, 490), (0, 1, 29, 165, 493))

# Number of rays
paramIneqs1_11[x][0]
5
# The inequalities in ps
paramIneqs1_11[x][2]
(p0, 2*p3 + p4, p2, p1)

# Sum Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY34}
{10: (0, 0),
 17: (0, 0),
 24: (0, 0),
 26: (p1 + p2 + p3, p1 + p2 + p3),
 27: (p3, 0),
 33: (0, p3)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY8}
{5: (p3, 0),
 7: (0, 0),
 9: (p1 + p2 + p3, p1 + p2 + p3),
 23: (0, 0),
 31: (0, p3),
 34: (0, 0)}

# The common non-zero remaining roots:
{k: rootEqualities1_11[x][k] for k in rootEqualities1_11[x].keys() if k not in sumY34 + sumY8}
{0: p0 + 2*p1 + 2*p2 + 3*p3 + p4,
 1: p4,
 2: p1 + p2 + p3,
 21: p1 + p2 + p3,
 29: 2*p1 + p2 + p3}

# The remaining different roots:
{k: rootDifferences1_11[x][k] for k in rootDifferences1_11[x].keys() if k not in sumY34 + sumY8}
{8: (0, p3), 28: (0, p3), 30: (p3, 0), 32: (p3, 0)}

#############################
# Cone (0, 1, 29, 165, 490) #
#############################

u =  allPairsOfTuples1_11[4][0]
print u
(0, 1, 29, 165, 490)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: 0, 26: p1 + p2 + p3, 27: p3, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p3, 7: 0, 9: p1 + p2 + p3, 23: 0, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][0] for k in rootDifferences1_11[x].keys()}
{5: p3, 8: 0, 27: p3, 28: 0, 30: p3, 31: 0, 32: p3, 33: 0}


#############################
# Cone (0, 1, 29, 165, 493) #
#############################

u =  allPairsOfTuples1_11[4][1]
print u
(0, 1, 29, 165, 493)

# # number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: 0, 26: p1 + p2 + p3, 27: 0, 33: p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: p1 + p2 + p3, 23: 0, 31: p3, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][1] for k in rootDifferences1_11[x].keys()}
{5: 0, 8: p3, 27: 0, 28: p3, 30: 0, 31: p3, 32: 0, 33: p3}



##########
# PAIR 5 #
##########

x = tuple(allPairsOfTuples1_11[5])
print x
((0, 4, 29, 309, 613), (0, 4, 29, 309, 636))

# Number of rays
paramIneqs1_11[x][0]
5
# The inequalities in ps
paramIneqs1_11[x][2]
(2*p3 + p4, p0, p2, p1)

# Sum Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY34}
{10: (p3, 0),
 17: (0, p3),
 24: (0, 0),
 26: (0, 0),
 27: (0, 0),
 33: (p0 + p1 + p2 + p3, p0 + p1 + p2 + p3)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k]) for k in sumY8}
{5: (p0 + p1 + p2 + p3, p0 + p1 + p2 + p3),
 7: (0, p3),
 9: (0, 0),
 23: (0, 0),
 31: (0, 0),
 34: (p3, 0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_11[x][k] for k in rootEqualities1_11[x].keys() if k not in sumY34 + sumY8}
{0: p4,
 4: p0 + 2*p1 + 2*p2 + p3,
 8: p0 + p1 + p2 + p3,
 29: p0 + 2*p1 + p2 + p3,
 32: p0 + p1 + p2 + p3}

# The remaining different roots:
{k: rootDifferences1_11[x][k] for k in rootDifferences1_11[x].keys() if k not in sumY34 + sumY8}
{3: (p3, 0), 13: (0, p3), 14: (p3, 0), 35: (0, p3)}

#############################
# Cone (0, 4, 29, 309, 613) #
#############################

u =  allPairsOfTuples1_11[5][0]
print u
(0, 4, 29, 309, 613)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p3, 17: 0, 24: 0, 26: 0, 27: 0, 33: p0 + p1 + p2 + p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p3, 7: 0, 9: 0, 23: 0, 31: 0, 34: p3}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][0] for k in rootDifferences1_11[x].keys()}
{3: p3, 7: 0, 10: p3, 13: 0, 14: p3, 17: 0, 34: p3, 35: 0}


#############################
# Cone (0, 4, 29, 309, 636) #
#############################

u =  allPairsOfTuples1_11[5][1]
print u
(0, 4, 29, 309, 636)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p3, 24: 0, 26: 0, 27: 0, 33: p0 + p1 + p2 + p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p3, 7: p3, 9: 0, 23: 0, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_11[x][k][1] for k in rootDifferences1_11[x].keys()}
{3: 0, 7: p3, 10: 0, 13: p3, 14: 0, 17: p3, 34: 0, 35: p3}



#########################
# Length 1 and 20 roots #
#########################

print len(len1_20)
24

testPolys_20 = {allTuplesForRelevantRoots[len1_20[k]][0]: Polyhedron(ieqs=IneqsFibersInaa2a3a4_lower_dim[allTuplesForRelevantRoots[len1_20[k]][0]]) for k in range(0,len(len1_20))}

allPolys1_20 = set(testPolys_20.values())
allTuplesPerPolys1_20 = {P : [u for u in testPolys_20 if testPolys_20[u]==P] for P in allPolys1_20}

# allTuplesPerPolys1_20.values()
# [[(1, 4, 204, 543, 707),
#   (1, 4, 204, 543, 703),
#   (1, 4, 204, 540, 702),
#   (1, 4, 204, 540, 706)],
#  [(4, 29, 309, 636, 711),
#   (4, 29, 309, 613, 710),
#   (4, 29, 309, 613, 712),
#   (4, 29, 309, 636, 713)],
#  [(0, 4, 172, 506, 700),
#   (0, 4, 172, 506, 701),
#   (0, 4, 172, 487, 694),
#   (0, 4, 172, 487, 697)],
#  [(0, 1, 158, 473, 689),
#   (0, 1, 158, 482, 690),
#   (0, 1, 158, 482, 693),
#   (0, 1, 158, 473, 688)],
#  [(1, 29, 238, 563, 709),
#   (1, 29, 238, 563, 705),
#   (1, 29, 238, 548, 708),
#   (1, 29, 238, 548, 704)],
#  [(0, 29, 165, 493, 699),
#   (0, 29, 165, 490, 695),
#   (0, 29, 165, 493, 696),
#   (0, 29, 165, 490, 698)]]

# # They come in quartets!
# set([len(x) for x in allTuplesPerPolys1_20.values()])
# {4}

# len(allTuplesPerPolys1_20.keys())
# 6


# We confirm that all quartets have the same inequalities, the same dictionary writing the scalars zs as linear expressions in the parameters ps.
all([all([all([bool(dictIneqpsRoots_lower_dim[x[0]][k]==dictIneqpsRoots_lower_dim[x[1]][k]), bool(dictIneqpsRoots_lower_dim[x[0]][k]==dictIneqpsRoots_lower_dim[x[2]][k]),bool(dictIneqpsRoots_lower_dim[x[0]][k]==dictIneqpsRoots_lower_dim[x[3]][k]) ]) for k in range(0,3)]) for x in allTuplesPerPolys1_20.values()])
True

# Check the roots for each quartet are different:
all([False == bool(dictIneqpsRoots_lower_dim[x[0]][3]==dictIneqpsRoots_lower_dim[x[1]][3]) for x in  allTuplesPerPolys1_20.values()])
True
all([False == bool(dictIneqpsRoots_lower_dim[x[0]][3]==dictIneqpsRoots_lower_dim[x[2]][3]) for x in  allTuplesPerPolys1_20.values()])
True
all([False == bool(dictIneqpsRoots_lower_dim[x[0]][3]==dictIneqpsRoots_lower_dim[x[3]][3]) for x in  allTuplesPerPolys1_20.values()])
True
all([False == bool(dictIneqpsRoots_lower_dim[x[1]][3]==dictIneqpsRoots_lower_dim[x[2]][3]) for x in  allTuplesPerPolys1_20.values()])
True
all([False == bool(dictIneqpsRoots_lower_dim[x[1]][3]==dictIneqpsRoots_lower_dim[x[3]][3]) for x in  allTuplesPerPolys1_20.values()])
True
all([False == bool(dictIneqpsRoots_lower_dim[x[2]][3]==dictIneqpsRoots_lower_dim[x[3]][3]) for x in  allTuplesPerPolys1_20.values()])
True


# The next dictionaries encode the similarities and difference between the cones with the same solution polyhedron in z1,...,z5.

rootDifferences1_20 = dict()
rootEqualities1_20 = dict()
paramIneqs1_20 = dict()

for x in allTuplesPerPolys1_20.values():
    print x
    # Write the different roots:
    notToCheck = [k for k in range(0,36) if dictIneqpsRoots_lower_dim[x[0]][3][k] == dictIneqpsRoots_lower_dim[x[1]][3][k] if dictIneqpsRoots_lower_dim[x[0]][3][k] == dictIneqpsRoots_lower_dim[x[2]][3][k] if dictIneqpsRoots_lower_dim[x[0]][3][k] == dictIneqpsRoots_lower_dim[x[3]][3][k]]
    
    rootDifferences1_20[tuple(x)] = {k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k],dictIneqpsRoots_lower_dim[x[2]][3][k],dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in range(0,36) if k not in notToCheck}
    rootEqualities1_20[tuple(x)] ={k: dictIneqpsRoots_lower_dim[x[0]][3][k] for k in range(0,36) if k in notToCheck if dictIneqpsRoots_lower_dim[x[0]][3][k] !=0}
    paramIneqs1_20[tuple(x)] = [dictIneqpsRoots_lower_dim[x[0]][k] for k in range(0,3)]


allPairsOfTuples1_20 = allTuplesPerPolys1_20.values()


##########
# PAIR 0 #
##########

x = tuple(allPairsOfTuples1_20[0])
print x
((1, 4, 204, 543, 707), (1, 4, 204, 543, 703), (1, 4, 204, 540, 702), (1, 4, 204, 540, 706))

# Number of rays
paramIneqs1_20[x][0]
5

# The inequalities in ps
paramIneqs1_20[x][2]
(2*p1, 2*p2, 2*p4, p3)

# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY34}
{10: (p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4),
 17: (0, 0, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4),
 24: (0, p0, 0, p0),
 26: (p0, 0, 0, p0),
 27: (p0, 0, p0, 0),
 33: (p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, 0, 0)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY8}
{5: (0, 0, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4),
 7: (p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4),
 9: (p0, 0, 0, p0),
 23: (p0, 0, p0, 0),
 31: (0, p0, 0, p0),
 34: (p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, 0, 0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_20[x][k] for k in rootEqualities1_20[x].keys() if k not in sumY34 + sumY8}
{1: p0 + p1 + 3*p2 + 2*p3 + 3*p4,
 3: p0 + p1 + p2 + p3 + p4,
 4: p0 + p1 + p2 + 2*p3 + 3*p4,
 13: p0 + p1 + p2 + p3 + p4}

# The remaining different roots:
{k: rootDifferences1_20[x][k] for k in rootDifferences1_20[x].keys() if k not in sumY34 + sumY8}
{2: (0, p0, p0, 0),
 6: (0, p0, p0, 0),
 8: (0, 0, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4),
 11: (0, p0, p0, 0),
 12: (p0, 0, 0, p0),
 14: (0, 0, p0 + p1 + p2 + p4, p0 + p1 + p2 + p4),
 15: (0, p0, p0, 0),
 16: (p0, 0, 0, p0),
 18: (p0, 0, p0, 0),
 19: (p0, 0, 0, p0),
 20: (0, p0, p0, 0),
 21: (0, p0, p0, 0),
 22: (p0, 0, 0, p0),
 25: (0, p0, 0, p0),
 28: (p0, 0, p0, 0),
 30: (0, p0, 0, p0),
 32: (p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, 0, 0),
 35: (p0 + p1 + p2 + p4, p0 + p1 + p2 + p4, 0, 0)}


##############################
# Cone (1, 4, 204, 543, 707) #
##############################

u =  allPairsOfTuples1_20[0][0]
print u
(1, 4, 204, 543, 707)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1 + p2 + p3 + p4,
 17: 0,
 24: 0,
 26: p0,
 27: p0,
 33: p0 + p1 + p2 + p4}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0 + p1 + p2 + p3 + p4, 9: p0, 23: p0, 31: 0, 34: p0 + p1 + p2 + p4}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][0] for k in rootDifferences1_20[x].keys()}
{2: 0,
 5: 0,
 6: 0,
 8: 0,
 9: p0,
 11: 0,
 12: p0,
 14: 0,
 15: 0,
 16: p0,
 17: 0,
 18: p0,
 19: p0,
 20: 0,
 21: 0,
 22: p0,
 23: p0,
 24: 0,
 25: 0,
 26: p0,
 27: p0,
 28: p0,
 30: 0,
 31: 0,
 32: p0 + p1 + p2 + p4,
 33: p0 + p1 + p2 + p4,
 34: p0 + p1 + p2 + p4,
 35: p0 + p1 + p2 + p4}



##############################
# Cone (1, 4, 204, 543, 703) #
##############################

u =  allPairsOfTuples1_20[0][1]
print u
(1, 4, 204, 543, 703)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots

# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1 + p2 + p3 + p4,
 17: 0,
 24: p0,
 26: 0,
 27: 0,
 33: p0 + p1 + p2 + p4}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0 + p1 + p2 + p3 + p4, 9: 0, 23: 0, 31: p0, 34: p0 + p1 + p2 + p4}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][1] for k in rootDifferences1_20[x].keys()}
{2: p0,
 5: 0,
 6: p0,
 8: 0,
 9: 0,
 11: p0,
 12: 0,
 14: 0,
 15: p0,
 16: 0,
 17: 0,
 18: 0,
 19: 0,
 20: p0,
 21: p0,
 22: 0,
 23: 0,
 24: p0,
 25: p0,
 26: 0,
 27: 0,
 28: 0,
 30: p0,
 31: p0,
 32: p0 + p1 + p2 + p4,
 33: p0 + p1 + p2 + p4,
 34: p0 + p1 + p2 + p4,
 35: p0 + p1 + p2 + p4}


##############################
# Cone (1, 4, 204, 540, 702) #
##############################

u =  allPairsOfTuples1_20[0][2]
print u
(1, 4, 204, 540, 702)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1 + p2 + p3 + p4,
 17: p0 + p1 + p2 + p4,
 24: 0,
 26: 0,
 27: p0,
 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p4, 7: p0 + p1 + p2 + p3 + p4, 9: 0, 23: p0, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][2] for k in rootDifferences1_20[x].keys()}
{2: p0,
 5: p0 + p1 + p2 + p4,
 6: p0,
 8: p0 + p1 + p2 + p4,
 9: 0,
 11: p0,
 12: 0,
 14: p0 + p1 + p2 + p4,
 15: p0,
 16: 0,
 17: p0 + p1 + p2 + p4,
 18: p0,
 19: 0,
 20: p0,
 21: p0,
 22: 0,
 23: p0,
 24: 0,
 25: 0,
 26: 0,
 27: p0,
 28: p0,
 30: 0,
 31: 0,
 32: 0,
 33: 0,
 34: 0,
 35: 0}




##############################
# Cone (1, 4, 204, 540, 706) #
##############################

u =  allPairsOfTuples1_20[0][3]
print u
(1, 4, 204, 540, 706)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1 + p2 + p3 + p4,
 17: p0 + p1 + p2 + p4,
 24: p0,
 26: p0,
 27: 0,
 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p4, 7: p0 + p1 + p2 + p3 + p4, 9: p0, 23: 0, 31: p0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][3] for k in rootDifferences1_20[x].keys()}
{2: 0,
 5: p0 + p1 + p2 + p4,
 6: 0,
 8: p0 + p1 + p2 + p4,
 9: p0,
 11: 0,
 12: p0,
 14: p0 + p1 + p2 + p4,
 15: 0,
 16: p0,
 17: p0 + p1 + p2 + p4,
 18: 0,
 19: p0,
 20: 0,
 21: 0,
 22: p0,
 23: 0,
 24: p0,
 25: p0,
 26: p0,
 27: 0,
 28: 0,
 30: p0,
 31: p0,
 32: 0,
 33: 0,
 34: 0,
 35: 0}







##########
# PAIR 1 #
##########

x = tuple(allPairsOfTuples1_20[1])
print x
((4, 29, 309, 636, 711), (4, 29, 309, 613, 710), (4, 29, 309, 613, 712), (4, 29, 309, 636, 713))

# Number of rays
paramIneqs1_20[x][0]

# The inequalities in ps
paramIneqs1_20[x][2]
(2*p0, p2, p4, p3)

# The Summand_Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY34}
{10: (0, p0 + p1, p0 + p1, 0),
 17: (p0 + p1, 0, 0, p0 + p1),
 24: (p1, 0, p1, 0),
 26: (p1, p1, 0, 0),
 27: (0, 0, p1, p1),
 33: (p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY8}
{5: (p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4),
 7: (p0 + p1, 0, 0, p0 + p1),
 9: (0, 0, p1, p1),
 23: (p1, 0, p1, 0),
 31: (p1, p1, 0, 0),
 34: (0, p0 + p1, p0 + p1, 0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_20[x][k] for k in rootEqualities1_20[x].keys() if k not in sumY34 + sumY8}
{4: p0 + p1 + p2 + 2*p3 + 2*p4,
 8: p0 + p1 + p2 + p3 + p4,
 29: p0 + p1 + p2 + 2*p3 + p4,
 32: p0 + p1 + p2 + p3 + p4}

# The remaining different roots:
{k: rootDifferences1_20[x][k] for k in rootDifferences1_20[x].keys() if k not in sumY34 + sumY8}
{2: (p1, p1, 0, 0),
 3: (0, p0 + p1, p0 + p1, 0),
 6: (0, p1, 0, p1),
 11: (0, p1, 0, p1),
 12: (p1, 0, p1, 0),
 13: (p0 + p1, 0, 0, p0 + p1),
 14: (0, p0 + p1, p0 + p1, 0),
 15: (p1, 0, p1, 0),
 16: (0, p1, 0, p1),
 18: (0, p1, 0, p1),
 19: (p1, 0, p1, 0),
 20: (p1, 0, p1, 0),
 21: (0, 0, p1, p1),
 22: (0, p1, 0, p1),
 25: (0, p1, 0, p1),
 28: (0, 0, p1, p1),
 30: (p1, p1, 0, 0),
 35: (p0 + p1, 0, 0, p0 + p1)}


###############################
# Cone (4, 29, 309, 636, 711) #
###############################

u =  allPairsOfTuples1_20[1][0]
print u
(4, 29, 309, 636, 711)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0 + p1, 24: p1, 26: p1, 27: 0, 33: p0 + p1 + p2 + p3 + p4}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p3 + p4, 7: p0 + p1, 9: 0, 23: p1, 31: p1, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][0] for k in rootDifferences1_20[x].keys()}
{2: p1,
 3: 0,
 6: 0,
 7: p0 + p1,
 9: 0,
 10: 0,
 11: 0,
 12: p1,
 13: p0 + p1,
 14: 0,
 15: p1,
 16: 0,
 17: p0 + p1,
 18: 0,
 19: p1,
 20: p1,
 21: 0,
 22: 0,
 23: p1,
 24: p1,
 25: 0,
 26: p1,
 27: 0,
 28: 0,
 30: p1,
 31: p1,
 34: 0,
 35: p0 + p1}



###############################
# Cone (4, 29, 309, 613, 710) #
###############################

u =  allPairsOfTuples1_20[1][1]
print u
(4, 29, 309, 613, 710)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1, 17: 0, 24: 0, 26: p1, 27: 0, 33: p0 + p1 + p2 + p3 + p4}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p3 + p4, 7: 0, 9: 0, 23: 0, 31: p1, 34: p0 + p1}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][1] for k in rootDifferences1_20[x].keys()}
{2: p1,
 3: p0 + p1,
 6: p1,
 7: 0,
 9: 0,
 10: p0 + p1,
 11: p1,
 12: 0,
 13: 0,
 14: p0 + p1,
 15: 0,
 16: p1,
 17: 0,
 18: p1,
 19: 0,
 20: 0,
 21: 0,
 22: p1,
 23: 0,
 24: 0,
 25: p1,
 26: p1,
 27: 0,
 28: 0,
 30: p1,
 31: p1,
 34: p0 + p1,
 35: 0}



###############################
# Cone (4, 29, 309, 613, 712) #
###############################

u =  allPairsOfTuples1_20[1][2]
print u
(4, 29, 309, 613, 712)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1, 17: 0, 24: p1, 26: 0, 27: p1, 33: p0 + p1 + p2 + p3 + p4}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p3 + p4, 7: 0, 9: p1, 23: p1, 31: 0, 34: p0 + p1}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][2] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: p0 + p1,
 6: 0,
 7: 0,
 9: p1,
 10: p0 + p1,
 11: 0,
 12: p1,
 13: 0,
 14: p0 + p1,
 15: p1,
 16: 0,
 17: 0,
 18: 0,
 19: p1,
 20: p1,
 21: p1,
 22: 0,
 23: p1,
 24: p1,
 25: 0,
 26: 0,
 27: p1,
 28: p1,
 30: 0,
 31: 0,
 34: p0 + p1,
 35: 0}


###############################
# Cone (4, 29, 309, 636, 713) #
###############################

u =  allPairsOfTuples1_20[1][3]
print u
(4, 29, 309, 636, 713)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0 + p1, 24: 0, 26: 0, 27: p1, 33: p0 + p1 + p2 + p3 + p4}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p2 + p3 + p4, 7: p0 + p1, 9: p1, 23: 0, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][3] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: 0,
 6: p1,
 7: p0 + p1,
 9: p1,
 10: 0,
 11: p1,
 12: 0,
 13: p0 + p1,
 14: 0,
 15: 0,
 16: p1,
 17: p0 + p1,
 18: p1,
 19: 0,
 20: 0,
 21: p1,
 22: p1,
 23: 0,
 24: 0,
 25: p1,
 26: 0,
 27: p1,
 28: p1,
 30: 0,
 31: 0,
 34: 0,
 35: p0 + p1}





##########
# PAIR 2 #
##########

x = tuple(allPairsOfTuples1_20[2])
print x
((0, 4, 172, 506, 700), (0, 4, 172, 506, 701), (0, 4, 172, 487, 694), (0, 4, 172, 487, 697))

# Number of rays
paramIneqs1_20[x][0]
5
# The inequalities in ps
paramIneqs1_20[x][2]
(p1, 2*p2, 2*p3, p4)

# The Summand_Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY34}
{10: (p0, 0, p0, 0),
 17: (0, p0, p0, 0),
 24: (0, p0, 0, p0),
 26: (p0 + p2 + p3, p0 + p2 + p3, 0, 0),
 27: (p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4),
 33: (0, 0, p0 + p2 + p3, p0 + p2 + p3)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY8}
{5: (p0 + p2 + p3, p0 + p2 + p3, 0, 0),
 7: (0, p0, 0, p0),
 9: (0, 0, p0 + p2 + p3, p0 + p2 + p3),
 23: (p0, 0, p0, 0),
 31: (p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4),
 34: (0, p0, p0, 0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_20[x][k] for k in rootEqualities1_20[x].keys() if k not in sumY34 + sumY8}
{0: p0 + p1 + 3*p2 + 3*p3 + 2*p4,
 4: p0 + p2 + 3*p3 + 2*p4,
 28: p0 + p2 + p3 + p4,
 30: p0 + p2 + p3 + p4}

# The remaining different roots:
{k: rootDifferences1_20[x][k] for k in rootDifferences1_20[x].keys() if k not in sumY34 + sumY8}
{2: (0, 0, p0 + p2 + p3, p0 + p2 + p3),
 3: (p0, 0, p0, 0),
 6: (0, p0, p0, 0),
 8: (p0 + p2 + p3, p0 + p2 + p3, 0, 0),
 11: (0, p0, p0, 0),
 12: (0, p0, p0, 0),
 13: (0, p0, 0, p0),
 14: (p0, 0, 0, p0),
 15: (p0, 0, 0, p0),
 16: (p0, 0, 0, p0),
 18: (0, p0, 0, p0),
 19: (0, p0, p0, 0),
 20: (p0, 0, 0, p0),
 21: (p0 + p2 + p3, p0 + p2 + p3, 0, 0),
 22: (p0, 0, 0, p0),
 25: (p0, 0, p0, 0),
 32: (0, 0, p0 + p2 + p3, p0 + p2 + p3),
 35: (p0, 0, 0, p0)}



##############################
# Cone (0, 4, 172, 506, 700) # 
##############################

u =  allPairsOfTuples1_20[2][0]
print u
(0, 4, 172, 506, 700)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0, 17: 0, 24: 0, 26: p0 + p2 + p3, 27: p0 + p2 + p3 + p4, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p2 + p3, 7: 0, 9: 0, 23: p0, 31: p0 + p2 + p3 + p4, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][0] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: p0,
 5: p0 + p2 + p3,
 6: 0,
 7: 0,
 8: p0 + p2 + p3,
 9: 0,
 10: p0,
 11: 0,
 12: 0,
 13: 0,
 14: p0,
 15: p0,
 16: p0,
 17: 0,
 18: 0,
 19: 0,
 20: p0,
 21: p0 + p2 + p3,
 22: p0,
 23: p0,
 24: 0,
 25: p0,
 26: p0 + p2 + p3,
 32: 0,
 33: 0,
 34: 0,
 35: p0}



##############################
# Cone (0, 4, 172, 506, 701) #
##############################

u =  allPairsOfTuples1_20[2][1]
print u
(0, 4, 172, 506, 701)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0, 24: p0, 26: p0 + p2 + p3, 27: p0 + p2 + p3 + p4, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p2 + p3, 7: p0, 9: 0, 23: 0, 31: p0 + p2 + p3 + p4, 34: p0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][1] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: 0,
 5: p0 + p2 + p3,
 6: p0,
 7: p0,
 8: p0 + p2 + p3,
 9: 0,
 10: 0,
 11: p0,
 12: p0,
 13: p0,
 14: 0,
 15: 0,
 16: 0,
 17: p0,
 18: p0,
 19: p0,
 20: 0,
 21: p0 + p2 + p3,
 22: 0,
 23: 0,
 24: p0,
 25: 0,
 26: p0 + p2 + p3,
 32: 0,
 33: 0,
 34: p0,
 35: 0}



##############################
# Cone (0, 4, 172, 487, 694) # 
##############################

u =  allPairsOfTuples1_20[2][2]
print u
(0, 4, 172, 487, 694)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0, 17: p0, 24: 0, 26: 0, 27: p0 + p2 + p3 + p4, 33: p0 + p2 + p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: p0 + p2 + p3, 23: p0, 31: p0 + p2 + p3 + p4, 34: p0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][2] for k in rootDifferences1_20[x].keys()}
{2: p0 + p2 + p3,
 3: p0,
 5: 0,
 6: p0,
 7: 0,
 8: 0,
 9: p0 + p2 + p3,
 10: p0,
 11: p0,
 12: p0,
 13: 0,
 14: 0,
 15: 0,
 16: 0,
 17: p0,
 18: 0,
 19: p0,
 20: 0,
 21: 0,
 22: 0,
 23: p0,
 24: 0,
 25: p0,
 26: 0,
 32: p0 + p2 + p3,
 33: p0 + p2 + p3,
 34: p0,
 35: 0}


##############################
# Cone (0, 4, 172, 487, 697) #
##############################

u =  allPairsOfTuples1_20[2][3]
print u
(0, 4, 172, 487, 697)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: p0, 26: 0, 27: p0 + p2 + p3 + p4, 33: p0 + p2 + p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0, 9: p0 + p2 + p3, 23: 0, 31: p0 + p2 + p3 + p4, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][3] for k in rootDifferences1_20[x].keys()}
{2: p0 + p2 + p3,
 3: 0,
 5: 0,
 6: 0,
 7: p0,
 8: 0,
 9: p0 + p2 + p3,
 10: 0,
 11: 0,
 12: 0,
 13: p0,
 14: p0,
 15: p0,
 16: p0,
 17: 0,
 18: p0,
 19: 0,
 20: p0,
 21: 0,
 22: p0,
 23: 0,
 24: p0,
 25: 0,
 26: 0,
 32: p0 + p2 + p3,
 33: p0 + p2 + p3,
 34: 0,
 35: p0}




##########
# PAIR 3 #
##########

x = tuple(allPairsOfTuples1_20[3])
print x
((0, 1, 158, 473, 689), (0, 1, 158, 482, 690), (0, 1, 158, 482, 693), (0, 1, 158, 473, 688))

# Number of rays
paramIneqs1_20[x][0]
5
# The inequalities in ps
paramIneqs1_20[x][2]
(p1, p3, 2*p2, p4)

# The Summand_Y34 roots
# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY34}
{10: (p0, 0, p0, 0),
 17: (p0 + p2, 0, 0, p0 + p2),
 24: (p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4),
 26: (0, p0 + p2, p0 + p2, 0),
 27: (0, p0, 0, p0),
 33: (p0, p0, 0, 0)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY8}
{5: (p0, p0, 0, 0),
 7: (0, p0, 0, p0),
 9: (p0 + p2, 0, 0, p0 + p2),
 23: (p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4, p0 + p2 + p4),
 31: (p0, 0, p0, 0),
 34: (0, p0 + p2, p0 + p2, 0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_20[x][k] for k in rootEqualities1_20[x].keys() if k not in sumY34 + sumY8}
{0: p0 + p1 + 3*p2 + p3 + 2*p4,
 1: p0 + 3*p2 + p3 + 2*p4,
 18: p0 + p2 + p4,
 25: p0 + p2 + p4}

# The remaining different roots:
{k: rootDifferences1_20[x][k] for k in rootDifferences1_20[x].keys() if k not in sumY34 + sumY8}
{2: (p0 + p2, 0, 0, p0 + p2),
 3: (0, p0, 0, p0),
 6: (0, 0, p0, p0),
 8: (0, 0, p0, p0),
 11: (p0, p0, 0, 0),
 12: (0, 0, p0, p0),
 13: (p0, 0, p0, 0),
 14: (p0 + p2, 0, 0, p0 + p2),
 15: (0, 0, p0, p0),
 16: (0, 0, p0, p0),
 19: (p0, p0, 0, 0),
 20: (p0, p0, 0, 0),
 21: (0, p0 + p2, p0 + p2, 0),
 22: (p0, p0, 0, 0),
 28: (p0, 0, p0, 0),
 30: (0, p0, 0, p0),
 32: (0, 0, p0, p0),
 35: (0, p0 + p2, p0 + p2, 0)}




##############################
# Cone (0, 1, 158, 473, 689) #
##############################

u =  allPairsOfTuples1_20[3][0]
print u
(0, 1, 158, 473, 689)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0, 17: p0 + p2, 24: p0 + p2 + p4, 26: 0, 27: 0, 33: p0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0, 7: 0, 9: p0 + p2, 23: p0 + p2 + p4, 31: p0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][0] for k in rootDifferences1_20[x].keys()}
{2: p0 + p2,
 3: 0,
 5: p0,
 6: 0,
 7: 0,
 8: 0,
 9: p0 + p2,
 10: p0,
 11: p0,
 12: 0,
 13: p0,
 14: p0 + p2,
 15: 0,
 16: 0,
 17: p0 + p2,
 19: p0,
 20: p0,
 21: 0,
 22: p0,
 26: 0,
 27: 0,
 28: p0,
 30: 0,
 31: p0,
 32: 0,
 33: p0,
 34: 0,
 35: 0}



##############################
# Cone (0, 1, 158, 482, 690) # 
##############################

u =  allPairsOfTuples1_20[3][1]
print u
(0, 1, 158, 482, 690)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: p0 + p2 + p4, 26: p0 + p2, 27: p0, 33: p0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0, 7: p0, 9: 0, 23: p0 + p2 + p4, 31: 0, 34: p0 + p2}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][1] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: p0,
 5: p0,
 6: 0,
 7: p0,
 8: 0,
 9: 0,
 10: 0,
 11: p0,
 12: 0,
 13: 0,
 14: 0,
 15: 0,
 16: 0,
 17: 0,
 19: p0,
 20: p0,
 21: p0 + p2,
 22: p0,
 26: p0 + p2,
 27: p0,
 28: 0,
 30: p0,
 31: 0,
 32: 0,
 33: p0,
 34: p0 + p2,
 35: p0 + p2}



##############################
# Cone (0, 1, 158, 482, 693) # 
##############################

u =  allPairsOfTuples1_20[3][2]
print u
(0, 1, 158, 482, 693)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0, 17: 0, 24: p0 + p2 + p4, 26: p0 + p2, 27: 0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: 0, 23: p0 + p2 + p4, 31: p0, 34: p0 + p2}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][2] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: 0,
 5: 0,
 6: p0,
 7: 0,
 8: p0,
 9: 0,
 10: p0,
 11: 0,
 12: p0,
 13: p0,
 14: 0,
 15: p0,
 16: p0,
 17: 0,
 19: 0,
 20: 0,
 21: p0 + p2,
 22: 0,
 26: p0 + p2,
 27: 0,
 28: p0,
 30: 0,
 31: p0,
 32: p0,
 33: 0,
 34: p0 + p2,
 35: p0 + p2}


##############################
# Cone (0, 1, 158, 473, 688) #
##############################

u =  allPairsOfTuples1_20[3][3]
print u
(0, 1, 158, 473, 688)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0 + p2, 24: p0 + p2 + p4, 26: 0, 27: p0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0, 9: p0 + p2, 23: p0 + p2 + p4, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][3] for k in rootDifferences1_20[x].keys()}
{2: p0 + p2,
 3: p0,
 5: 0,
 6: p0,
 7: p0,
 8: p0,
 9: p0 + p2,
 10: 0,
 11: 0,
 12: p0,
 13: 0,
 14: p0 + p2,
 15: p0,
 16: p0,
 17: p0 + p2,
 19: 0,
 20: 0,
 21: 0,
 22: 0,
 26: 0,
 27: p0,
 28: 0,
 30: p0,
 31: 0,
 32: p0,
 33: 0,
 34: 0,
 35: 0}





##########
# PAIR 4 #
##########

x = tuple(allPairsOfTuples1_20[4])
print x
((1, 29, 238, 563, 709), (1, 29, 238, 563, 705), (1, 29, 238, 548, 708), (1, 29, 238, 548, 704))

# Number of rays
paramIneqs1_20[x][0]
5
# The inequalities in ps
paramIneqs1_20[x][2]
(2*p1, 2*p3, p4, p2)

# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY34}
{10: (p0 + p1 + p3, p0 + p1 + p3, 0, 0),
 17: (p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4),
 24: (p0, 0, p0, 0),
 26: (0, p0, 0, p0),
 27: (p0, 0, 0, p0),
 33: (0, 0, p0 + p1 + p3, p0 + p1 + p3)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY8}
{5: (p0 + p1 + p3, p0 + p1 + p3, 0, 0),
 7: (0, 0, p0 + p1 + p3, p0 + p1 + p3),
 9: (p0, 0, p0, 0),
 23: (0, p0, 0, p0),
 31: (p0, 0, 0, p0),
 34: (p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4,
  p0 + p1 + p2 + p3 + p4)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_20[x][k] for k in rootEqualities1_20[x].keys() if k not in sumY34 + sumY8}
{1: p0 + p1 + 2*p2 + 3*p3 + 2*p4,
 14: p0 + p1 + p2 + p3 + p4,
 29: p0 + p1 + 2*p2 + p3 + p4,
 35: p0 + p1 + p2 + p3 + p4}

# The remaining different roots:
{k: rootDifferences1_20[x][k] for k in rootDifferences1_20[x].keys() if k not in sumY34 + sumY8}
{2: (0, p0, 0, p0),
 3: (0, 0, p0 + p1 + p3, p0 + p1 + p3),
 6: (p0, 0, 0, p0),
 8: (0, 0, p0 + p1 + p3, p0 + p1 + p3),
 11: (0, p0, p0, 0),
 12: (0, p0, p0, 0),
 13: (p0 + p1 + p3, p0 + p1 + p3, 0, 0),
 15: (p0, 0, 0, p0),
 16: (0, p0, p0, 0),
 18: (0, p0, 0, p0),
 19: (p0, 0, 0, p0),
 20: (0, p0, p0, 0),
 21: (p0, 0, p0, 0),
 22: (p0, 0, 0, p0),
 25: (p0, 0, p0, 0),
 28: (0, p0, p0, 0),
 30: (0, p0, p0, 0),
 32: (p0 + p1 + p3, p0 + p1 + p3, 0, 0)}





###############################
# Cone (1, 29, 238, 563, 709) #
###############################

u =  allPairsOfTuples1_20[4][0]
print u
(1, 29, 238, 563, 709)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1 + p3, 17: p0 + p1 + p2 + p3 + p4, 24: p0, 26: 0, 27: p0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p3, 7: 0, 9: p0, 23: 0, 31: p0, 34: p0 + p1 + p2 + p3 + p4}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][0] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: 0,
 5: p0 + p1 + p3,
 6: p0,
 7: 0,
 8: 0,
 9: p0,
 10: p0 + p1 + p3,
 11: 0,
 12: 0,
 13: p0 + p1 + p3,
 15: p0,
 16: 0,
 18: 0,
 19: p0,
 20: 0,
 21: p0,
 22: p0,
 23: 0,
 24: p0,
 25: p0,
 26: 0,
 27: p0,
 28: 0,
 30: 0,
 31: p0,
 32: p0 + p1 + p3,
 33: 0}



###############################
# Cone (1, 29, 238, 563, 705) #
###############################

u =  allPairsOfTuples1_20[4][1]
print u
(1, 29, 238, 563, 705)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0 + p1 + p3, 17: p0 + p1 + p2 + p3 + p4, 24: 0, 26: p0, 27: 0, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p1 + p3, 7: 0, 9: 0, 23: p0, 31: 0, 34: p0 + p1 + p2 + p3 + p4}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][1] for k in rootDifferences1_20[x].keys()}
{2: p0,
 3: 0,
 5: p0 + p1 + p3,
 6: 0,
 7: 0,
 8: 0,
 9: 0,
 10: p0 + p1 + p3,
 11: p0,
 12: p0,
 13: p0 + p1 + p3,
 15: 0,
 16: p0,
 18: p0,
 19: 0,
 20: p0,
 21: 0,
 22: 0,
 23: p0,
 24: 0,
 25: 0,
 26: p0,
 27: 0,
 28: p0,
 30: p0,
 31: 0,
 32: p0 + p1 + p3,
 33: 0}



###############################
# Cone (1, 29, 238, 548, 708) #
###############################

u =  allPairsOfTuples1_20[4][2]
print u
(1, 29, 238, 548, 708)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0 + p1 + p2 + p3 + p4, 24: p0, 26: 0, 27: 0, 33: p0 + p1 + p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0 + p1 + p3, 9: p0, 23: 0, 31: 0, 34: p0 + p1 + p2 + p3 + p4}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][2] for k in rootDifferences1_20[x].keys()}
{2: 0,
 3: p0 + p1 + p3,
 5: 0,
 6: 0,
 7: p0 + p1 + p3,
 8: p0 + p1 + p3,
 9: p0,
 10: 0,
 11: p0,
 12: p0,
 13: 0,
 15: 0,
 16: p0,
 18: 0,
 19: 0,
 20: p0,
 21: p0,
 22: 0,
 23: 0,
 24: p0,
 25: p0,
 26: 0,
 27: 0,
 28: p0,
 30: p0,
 31: 0,
 32: 0,
 33: p0 + p1 + p3}


###############################
# Cone (1, 29, 238, 548, 704) #
###############################

u =  allPairsOfTuples1_20[4][3]
print u
(1, 29, 238, 548, 704)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0 + p1 + p2 + p3 + p4, 24: 0, 26: p0, 27: p0, 33: p0 + p1 + p3}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0 + p1 + p3, 9: 0, 23: p0, 31: p0, 34: p0 + p1 + p2 + p3 + p4}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][3] for k in rootDifferences1_20[x].keys()}
{2: p0,
 3: p0 + p1 + p3,
 5: 0,
 6: p0,
 7: p0 + p1 + p3,
 8: p0 + p1 + p3,
 9: 0,
 10: 0,
 11: 0,
 12: 0,
 13: 0,
 15: p0,
 16: 0,
 18: p0,
 19: p0,
 20: 0,
 21: 0,
 22: p0,
 23: p0,
 24: 0,
 25: 0,
 26: p0,
 27: p0,
 28: 0,
 30: 0,
 31: p0,
 32: 0,
 33: p0 + p1 + p3}





##########
# PAIR 5 #
##########

x = tuple(allPairsOfTuples1_20[5])
print x
((0, 29, 165, 493, 699), (0, 29, 165, 490, 695), (0, 29, 165, 493, 696), (0, 29, 165, 490, 698))

# Number of rays
paramIneqs1_20[x][0]
5
# The inequalities in ps
paramIneqs1_20[x][2]
(p1, 2*p2, p4, p3)

# The Summand_Y34 roots:
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY34}
{10: (p0, 0, 0, p0),
 17: (p0, p0, 0, 0),
 24: (0, 0, p0, p0),
 26: (p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4),
 27: (0, p0 + p2, 0, p0 + p2),
 33: (p0 + p2, 0, p0 + p2, 0)}

# The Summand_Y8 roots
{k:(dictIneqpsRoots_lower_dim[x[0]][3][k], dictIneqpsRoots_lower_dim[x[1]][3][k], dictIneqpsRoots_lower_dim[x[2]][3][k], dictIneqpsRoots_lower_dim[x[3]][3][k]) for k in sumY8}
{5: (0, p0 + p2, 0, p0 + p2),
 7: (p0, 0, 0, p0),
 9: (p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4,
  p0 + p2 + p3 + p4),
 23: (p0, p0, 0, 0),
 31: (p0 + p2, 0, p0 + p2, 0),
 34: (0, 0, p0, p0)}
 
# The common non-zero remaining roots:
{k: rootEqualities1_20[x][k] for k in rootEqualities1_20[x].keys() if k not in sumY34 + sumY8}
{0: p0 + p1 + 3*p2 + 2*p3 + 2*p4,
 2: p0 + p2 + p3 + p4,
 21: p0 + p2 + p3 + p4,
 29: p0 + p2 + 2*p3 + p4}

# The remaining different roots:
{k: rootDifferences1_20[x][k] for k in rootDifferences1_20[x].keys() if k not in sumY34 + sumY8}
{3: (0, p0, p0, 0),
 6: (0, p0, p0, 0),
 8: (p0 + p2, 0, p0 + p2, 0),
 11: (p0, 0, 0, p0),
 12: (0, p0, p0, 0),
 13: (0, p0, p0, 0),
 14: (0, 0, p0, p0),
 15: (p0, 0, 0, p0),
 16: (p0, 0, 0, p0),
 18: (0, 0, p0, p0),
 19: (p0, 0, 0, p0),
 20: (0, p0, p0, 0),
 22: (0, p0, p0, 0),
 25: (p0, p0, 0, 0),
 28: (p0 + p2, 0, p0 + p2, 0),
 30: (0, p0 + p2, 0, p0 + p2),
 32: (0, p0 + p2, 0, p0 + p2),
 35: (p0, p0, 0, 0)}



###############################
# Cone (0, 29, 165, 493, 699) #
###############################

u =  allPairsOfTuples1_20[5][0]
print u
(0, 29, 165, 493, 699)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0, 17: p0, 24: 0, 26: p0 + p2 + p3 + p4, 27: 0, 33: p0 + p2}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: p0, 9: p0 + p2 + p3 + p4, 23: p0, 31: p0 + p2, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][0] for k in rootDifferences1_20[x].keys()}
{3: 0,
 5: 0,
 6: 0,
 7: p0,
 8: p0 + p2,
 10: p0,
 11: p0,
 12: 0,
 13: 0,
 14: 0,
 15: p0,
 16: p0,
 17: p0,
 18: 0,
 19: p0,
 20: 0,
 22: 0,
 23: p0,
 24: 0,
 25: p0,
 27: 0,
 28: p0 + p2,
 30: 0,
 31: p0 + p2,
 32: 0,
 33: p0 + p2,
 34: 0,
 35: p0}



###############################
# Cone (0, 29, 165, 490, 695) #
###############################

u =  allPairsOfTuples1_20[5][1]
print u
(0, 29, 165, 490, 695)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: p0, 24: 0, 26: p0 + p2 + p3 + p4, 27: p0 + p2, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p2, 7: 0, 9: p0 + p2 + p3 + p4, 23: p0, 31: 0, 34: 0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][1] for k in rootDifferences1_20[x].keys()}
{3: p0,
 5: p0 + p2,
 6: p0,
 7: 0,
 8: 0,
 10: 0,
 11: 0,
 12: p0,
 13: p0,
 14: 0,
 15: 0,
 16: 0,
 17: p0,
 18: 0,
 19: 0,
 20: p0,
 22: p0,
 23: p0,
 24: 0,
 25: p0,
 27: p0 + p2,
 28: 0,
 30: p0 + p2,
 31: 0,
 32: p0 + p2,
 33: 0,
 34: 0,
 35: p0}



###############################
# Cone (0, 29, 165, 493, 696) #
###############################

u =  allPairsOfTuples1_20[5][2]
print u
(0, 29, 165, 493, 696)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: 0, 17: 0, 24: p0, 26: p0 + p2 + p3 + p4, 27: 0, 33: p0 + p2}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: 0, 7: 0, 9: p0 + p2 + p3 + p4, 23: 0, 31: p0 + p2, 34: p0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][2] for k in rootDifferences1_20[x].keys()}
{3: p0,
 5: 0,
 6: p0,
 7: 0,
 8: p0 + p2,
 10: 0,
 11: 0,
 12: p0,
 13: p0,
 14: p0,
 15: 0,
 16: 0,
 17: 0,
 18: p0,
 19: 0,
 20: p0,
 22: p0,
 23: 0,
 24: p0,
 25: 0,
 27: 0,
 28: p0 + p2,
 30: 0,
 31: p0 + p2,
 32: 0,
 33: p0 + p2,
 34: p0,
 35: 0}


###############################
# Cone (0, 29, 165, 490, 698) #
###############################

u =  allPairsOfTuples1_20[5][3]
print u
(0, 29, 165, 490, 698)

# # Number of rays
# dictIneqpsRoots_lower_dim[u][0]
# # The inequalities in ps
# dictIneqpsRoots_lower_dim[u][2]
# # The coefficients of the relevant roots:
# relevantRoots = {k:dictIneqpsRoots_lower_dim[u][3][k] for k in rootValuesaa2a3a4[u].keys()}
# relevantRoots


# The Summand_Y34 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY34}
{10: p0, 17: 0, 24: p0, 26: p0 + p2 + p3 + p4, 27: p0 + p2, 33: 0}

# The Summand_Y8 roots:
{k:dictIneqpsRoots_lower_dim[u][3][k] for k in sumY8}
{5: p0 + p2, 7: p0, 9: p0 + p2 + p3 + p4, 23: 0, 31: 0, 34: p0}

# The coefficients of the relevant different roots:
{k:rootDifferences1_20[x][k][3] for k in rootDifferences1_20[x].keys()}
{3: 0,
 5: p0 + p2,
 6: 0,
 7: p0,
 8: 0,
 10: p0,
 11: p0,
 12: 0,
 13: 0,
 14: p0,
 15: p0,
 16: p0,
 17: 0,
 18: p0,
 19: p0,
 20: 0,
 22: 0,
 23: 0,
 24: p0,
 25: 0,
 27: p0 + p2,
 28: 0,
 30: p0 + p2,
 31: 0,
 32: p0 + p2,
 33: 0,
 34: p0,
 35: 0}