



"""simple docstring"""







import functools
from typing import Any

def     UpperCamelCase_   (   lowerCAmelCase__				:    str   ,       lowerCAmelCase__				:    list[str]						)						->       bool:

  """simple docstring"""




  if not isinstance(lowerCAmelCase__   ,       lowerCAmelCase__						) or len(lowerCAmelCase__						) == 0:
    raise ValueError('the string should be not empty string'						)

  if not isinstance(lowerCAmelCase__   ,       lowerCAmelCase__						) or not all(
      isinstance(lowerCAmelCase__   ,       lowerCAmelCase__						) and len(lowerCAmelCase__						) > 0 for item in words						):
    raise ValueError('the words should be a list of non-empty strings'						)

  # Build trie
  lowerCAmelCase_       :    dict[str, Any]								=			{}
  lowerCAmelCase_       :    Tuple								=			"""WORD_KEEPER"""

  for word in words:
    lowerCAmelCase_       :    List[str]								=			trie
    for c in word:
      if c not in trie_node:
        lowerCAmelCase_       :    Tuple								=			{}

      lowerCAmelCase_       :    Dict								=			trie_node[c]

    lowerCAmelCase_       :    Any								=			True

  lowerCAmelCase_       :    str								=			len(lowerCAmelCase__						)

  # Dynamic programming method
  @functools.cache
  def is_breakable(lowerCAmelCase__				:    int						) -> bool:
    if index == len_string:
      return True

    lowerCAmelCase_       :    Optional[int]								=			trie
    for i in range(lowerCAmelCase__   ,       lowerCAmelCase__						):
      lowerCAmelCase_       :    Tuple								=			trie_node.get(string[i]   ,       lowerCAmelCase__						)

      if trie_node is None:
        return False

      if trie_node.get(lowerCAmelCase__   ,       lowerCAmelCase__						) and is_breakable(i + 1						):
        return True

    return False

  return is_breakable(0						)


if __name__ == "__main__":
     import doctest

     doctest.testmod()










"""simple docstring"""







from __future__ import annotations

from fractions import Fraction
from math import gcd, sqrt

def     UpperCamelCase_   (   lowerCAmelCase__				:    int						)						->       bool:

  """simple docstring"""




  lowerCAmelCase_       :    int								=			int(number**0.5						)
  return number == sq * sq

def     UpperCamelCase_   (   lowerCAmelCase__				:    int   ,       lowerCAmelCase__				:    int   ,       lowerCAmelCase__				:    int   ,       lowerCAmelCase__				:    int   ,       lowerCAmelCase__				:    int   ,       lowerCAmelCase__				:    int						)						->       tuple[int, int]:

  """simple docstring"""




  lowerCAmelCase_       :    int								=			x_num * y_den * z_den + y_num * x_den * z_den + z_num * x_den * y_den
  lowerCAmelCase_       :    int								=			x_den * y_den * z_den
  lowerCAmelCase_       :    int								=			gcd(lowerCAmelCase__   ,       lowerCAmelCase__						)
  top //= hcf
  bottom //= hcf
  return top, bottom

def     UpperCamelCase_   (   lowerCAmelCase__				:    int = 35						)						->       int:

  """simple docstring"""




  lowerCAmelCase_       :    set								=			set()
  lowerCAmelCase_       :    int
  lowerCAmelCase_       :    Fraction								=			Fraction(0						)
  lowerCAmelCase_       :    tuple[int, int]

  for x_num in range(1   ,       order + 1						):
    for x_den in range(x_num + 1   ,       order + 1						):
      for y_num in range(1   ,       order + 1						):
        for y_den in range(y_num + 1   ,       order + 1						):
          # n=1
          lowerCAmelCase_       :    str								=			x_num * y_den + x_den * y_num
          lowerCAmelCase_       :    int								=			x_den * y_den
          lowerCAmelCase_       :    int								=			gcd(lowerCAmelCase__   ,       lowerCAmelCase__						)
          z_num //= hcf
          z_den //= hcf
          if 0 < z_num < z_den <= order:
            lowerCAmelCase_       :    List[str]								=			add_three(
                lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__						)
            unique_s.add(lowerCAmelCase__						)

          # n=2
          lowerCAmelCase_       :    Optional[int]								=			(
              x_num * x_num * y_den * y_den + x_den * x_den * y_num * y_num
          )
          lowerCAmelCase_       :    Dict								=			x_den * x_den * y_den * y_den
          if is_sq(lowerCAmelCase__						) and is_sq(lowerCAmelCase__						):
            lowerCAmelCase_       :    Optional[int]								=			int(sqrt(lowerCAmelCase__						)						)
            lowerCAmelCase_       :    List[str]								=			int(sqrt(lowerCAmelCase__						)						)
            lowerCAmelCase_       :    Union[str, Any]								=			gcd(lowerCAmelCase__   ,       lowerCAmelCase__						)
            z_num //= hcf
            z_den //= hcf
            if 0 < z_num < z_den <= order:
              lowerCAmelCase_       :    Union[str, Any]								=			add_three(
                  lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__						)
              unique_s.add(lowerCAmelCase__						)

                    # n=-1
          lowerCAmelCase_       :    Dict								=			x_num * y_num
          lowerCAmelCase_       :    Optional[int]								=			x_den * y_num + x_num * y_den
          lowerCAmelCase_       :    Any								=			gcd(lowerCAmelCase__   ,       lowerCAmelCase__						)
          z_num //= hcf
          z_den //= hcf
          if 0 < z_num < z_den <= order:
            lowerCAmelCase_       :    Tuple								=			add_three(
                lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__						)
            unique_s.add(lowerCAmelCase__						)

          # n=2
          lowerCAmelCase_       :    List[str]								=			x_num * x_num * y_num * y_num
          lowerCAmelCase_       :    Optional[Any]								=			(
              x_den * x_den * y_num * y_num + x_num * x_num * y_den * y_den
          )
          if is_sq(lowerCAmelCase__						) and is_sq(lowerCAmelCase__						):
            lowerCAmelCase_       :    Tuple								=			int(sqrt(lowerCAmelCase__						)						)
            lowerCAmelCase_       :    Optional[Any]								=			int(sqrt(lowerCAmelCase__						)						)
            lowerCAmelCase_       :    Optional[int]								=			gcd(lowerCAmelCase__   ,       lowerCAmelCase__						)
            z_num //= hcf
            z_den //= hcf
            if 0 < z_num < z_den <= order:
              lowerCAmelCase_       :    Any								=			add_three(
                  lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__   ,       lowerCAmelCase__						)
              unique_s.add(lowerCAmelCase__						)

  for num, den in unique_s:
    total += Fraction(lowerCAmelCase__   ,       lowerCAmelCase__						)

  return total.denominator + total.numerator


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





