
from collections import defaultdict
from math import ceil, sqrt




def        __UpperCamelCase					(    lowerCAmelCase__							:    Any = 1_0_0_0_0_0_0  ,       lowerCAmelCase__							:    Optional[int] = 1_0  ):
 __a			:  int		    =  defaultdict(SCREAMING_SNAKE_CASE_  )

 for outer_width in range(3  ,       (t_limit // 4) + 2  ):
  if outer_width * outer_width > t_limit:
   __a			:  Union[str, Any]		    =  max(
       ceil(sqrt(outer_width * outer_width - t_limit  )  )  ,       1  )
  else:
   __a			:  List[Any]		    =  1

  hole_width_lower_bound += (outer_width - hole_width_lower_bound) % 2

  for hole_width in range(SCREAMING_SNAKE_CASE_  ,       outer_width - 1  ,       2  ):
   count[outer_width * outer_width - hole_width * hole_width] += 1

 return sum(1 for n in count.values() if 1 <= n <= 1_0  )


if __name__ == "__main__":
      print(F"""{solution() = }""")


def        __UpperCamelCase					(    lowerCAmelCase__							:    int = 1_0_0_0  ):
 __a			:  int		    =  -1
 __a			:  Any		    =  0
 for a in range(1  ,       n // 3  ):
  # Solving the two equations a**2+b**2=c**2 and a+b+c=N eliminating c
  __a			:  List[Any]		    =  (n * n - 2 * a * n) // (2 * n - 2 * a)
  __a			:  Union[str, Any]		    =  n - a - b
  if c * c == (a * a + b * b):
   __a			:  Union[str, Any]		    =  a * b * c
   if candidate >= product:
    __a			:  List[Any]		    =  candidate
 return product


if __name__ == "__main__":
      print(F"""{solution() = }""")
