python3 -c 'import math,itertools
s=math.sqrt(3);b=math.sqrt(2)
def f(j,m):return math.sqrt(1/3)*(math.sin((j+1)*m*math.pi/6)-math.sin(j*m*math.pi/6))
def C(r,p,q,t):return sum(f(j,r)*f(j,p)*f(j,q)*f(j,t) for j in range(6))
def D(r,p,q,t,u,v):return sum(f(j,r)*f(j,p)*f(j,q)*f(j,t)*f(j,u)*f(j,v) for j in range(6))
w=[0,math.sqrt(2-s),1,math.sqrt(2),math.sqrt(3),math.sqrt(2+s)];a=[0,0,2,math.sqrt(3),2,0];M=[2,3,4];lab={2:(1,0,0),3:(0,1,0),4:(0,0,1)}
def add(d,k,v):d[k]=d.get(k,0)+v
def ng(k):return(-k[0],-k[1],-k[2])
def fold(f):
 g={}
 for k,v in f.items():
  kk=k
  if k[0]<0 or (k[0]==0 and k[1]<0) or (k[0]==0 and k[1]==0 and k[2]<0):kk=ng(k)
  add(g,kk,v)
 return g
def ev(k):return k[0]+k[1]*b+k[2]*s
def conv(f,g):
 h={}
 for k,v in f.items():
  for l,u in g.items():add(h,(k[0]+l[0],k[1]+l[1],k[2]+l[2]),v*u)
 return h
q3=[]
z=(0,0,0)
for p in range(6):
 fd={}
 for tr in itertools.product(M,repeat=3):
  cc=C(p,*tr)*math.prod(a[i] for i in tr)/6
  for sg in itertools.product([1,-1],repeat=3):
   k=tuple(sg[d]*lab[tr[d]][d] for d in range(3))
   add(fd,k,cc/8)
 fc=fold(fd);q={}
 for k,F in fc.items():
  if k==z:Q=F/w[p]**2
  else:
   vv=ev(k);Q=0 if abs(vv-w[p])<1e-10 else F/(w[p]**2-vv**2)
  add(q,k,Q)
 q3.append(q)
q1f={}
for m in M:
 e=lab[m];add(q1f,e,a[m]/2);add(q1f,ng(e),a[m]/2)
for r in [1,3,5]:
 Fd={}
 for tr in itertools.product(M,repeat=5):
  cc=-D(r,*tr)*math.prod(a[i] for i in tr)/120
  for sg in itertools.product([1,-1],repeat=5):
   k=tuple(sg[d]*lab[tr[d]][d] for d in range(3))
   add(Fd,k,cc/32)
 Fc={}
 for p in range(1,6):
  qf={}
  for k,Q in q3[p].items():
   if k==z:add(qf,k,Q)
   else:add(qf,k,Q/2);add(qf,ng(k),Q/2)
  for q in M:
   for t in M:
    e=lab[q];g={};add(g,e,a[q]/2);add(g,ng(e),a[q]/2)
    e=lab[t];h={};add(h,e,a[t]/2);add(h,ng(e),a[t]/2)
    for k,v in conv(conv(qf,g),h).items():add(Fc,k,0.5*C(r,p,q,t)*v)
 K=(2,1,2);cF=fold(Fd).get(K,0)+fold(Fc).get(K,0)
 print(r,cF/(w[r]**2-ev(K)**2))'