#!/usr/bin/env python3
from functools import lru_cache
V=[0,11,10,9,32,30,28,14,17,22,41,45,48]
O=[0]*13;O[8]=7
E=[(1,4),(1,8),(2,5),(2,8),(3,6),(3,7),(7,8),(8,9),(8,10),(8,11),(8,12)]
D=[[]for _ in range(13)]
for k,(a,b)in enumerate(E):D[a].append((k,b))
M=4095
def t(S,A):
 if S==0:return(1,0,M)
 G=[[]for _ in range(13)]
 for x in(1,2,3):
  if(S>>(x-1))%2:G[0].append(x)
 for k,(a,b)in enumerate(E):
  if(S>>(a-1))%2 and(S>>(b-1))%2 and(A>>k)%2:G[a].append(b)
 m=max((V[i],-i)for i in range(1,13)if(S>>(i-1))%2);w=-m[1]
 def R(ban):
  x={0};q=[0]
  while q:
   u=q.pop()
   for v in G[u]:
    if v!=ban and v not in x:x.add(v);q.append(v)
  return x
 sec=max([0,*[V[i]for i in range(1,13)if i!=w and(S>>(i-1))%2]])
 p=0
 for j in range(1,13):
  if(S>>(j-1))%2:
   z=V[j]-sec if j==w else(7 if w not in R(j)else 0)
  else:z=O[j]
  if z>0:p|=1<<(j-1)
 return(1<<S,p,M^p)
@lru_cache(None)
def f(i,S,A):
 if i>12:return t(S,A)
 inv=i<4
 if not inv:
  for k,(a,b)in enumerate(E):
   if b==i and a<i and(S>>(a-1))%2 and(A>>k)%2:inv=1;break
 if not inv:return f(-~i,S,A)
 sp=0;po=0;az=M
 for m in range(1<<len(D[i])):
  A2=A
  for q,(k,b)in enumerate(D[i]):
   if(m>>q)%2:A2|=1<<k
  x=f(-~i,S|(1<<(i-1)),A2);sp|=x[0];po|=x[1];az=int.__and__(az,x[2])
 x=f(-~i,S,A)
 if i==8:
  if S==2 or S==6:ac=[(sp,po,az),x]
  else:ac=[x]
 else:
  ac=[(sp,po,az)]
  if(az>>(i-1))%2:ac.append(x)
 rp=0;rpo=0;raz=0
 for a in ac:rp|=a[0];rpo|=a[1];raz|=a[2]
 return(rp,rpo,raz)
r=f(1,0,0)
print(bin(r[0]).count('1'))
print([i for i in range(4096)if(r[0]>>i)%2])
print(f.cache_info())