"""simple docstring"""
from __future__ import annotations







def 						__a   (	_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE = None		,     _SCREAMING_SNAKE_CASE = None		,     _SCREAMING_SNAKE_CASE = False		,     )	->tuple[int, float, str]:
  a__:     List[Any]										=	cipher_alphabet or [chr(_SCREAMING_SNAKE_CASE   ) for i in range(97		,     123   )]

  # If the argument is None or the user provided an empty dictionary
  if not frequencies_dict:
    # Frequencies of letters in the english language (how much they show up)
    a__:     List[Any]										=	{
        'a': 0.08_497,
        'b': 0.01_492,
        'c': 0.02_202,
        'd': 0.04_253,
        'e': 0.11_162,
        'f': 0.02_228,
        'g': 0.02_015,
        'h': 0.06_094,
        'i': 0.07_546,
        'j': 0.00_153,
        'k': 0.01_292,
        'l': 0.04_025,
        'm': 0.02_406,
        'n': 0.06_749,
        'o': 0.07_507,
        'p': 0.01_929,
        'q': 0.00_095,
        'r': 0.07_587,
        's': 0.06_327,
        't': 0.09_356,
        'u': 0.02_758,
        'v': 0.00_978,
        'w': 0.02_560,
        'x': 0.00_150,
        'y': 0.01_994,
        'z': 0.00_077,
    }
  else:
    # Custom frequencies dictionary
    a__:     Dict										=	frequencies_dict

  if not case_sensitive:
    a__:     Any										=	ciphertext.lower()

  # Chi squared statistic values
  a__:     dict[int, tuple[float, str]]										=	{}

  # cycle through all of the shifts
  for shift in range(len(_SCREAMING_SNAKE_CASE   )   ):
    a__:     Optional[Any]										=	''

    # decrypt the message with the shift
    for letter in ciphertext:
      try:
        # Try to index the letter in the alphabet
        a__:     Optional[Any]										=	(alphabet_letters.index(letter.lower()   ) - shift) % len(
            _SCREAMING_SNAKE_CASE   )
        decrypted_with_shift += (
            alphabet_letters[new_key].upper()
            if case_sensitive and letter.isupper()
            else alphabet_letters[new_key]
        )
      except ValueError:
        # Append the character if it isn't in the alphabet
        decrypted_with_shift += letter

    a__:     Union[str, Any]										=	0.0

    # Loop through each letter in the decoded message with the shift
    for letter in decrypted_with_shift:
      if case_sensitive:
        a__:     int										=	letter.lower()
        if letter in frequencies:
          # Get the amount of times the letter occurs in the message
          a__:     Optional[Any]										=	decrypted_with_shift.lower().count(_SCREAMING_SNAKE_CASE   )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          a__:     Any										=	frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          a__:     Optional[Any]										=	((occurrences - expected) ** 2) / expected

          # Add the margin of error to the total chi squared statistic
          chi_squared_statistic += chi_letter_value
      else:
        if letter.lower() in frequencies:
          # Get the amount of times the letter occurs in the message
          a__:     Union[str, Any]										=	decrypted_with_shift.count(_SCREAMING_SNAKE_CASE   )

          # Get the excepcted amount of times the letter should appear based
          # on letter frequencies
          a__:     Optional[int]										=	frequencies[letter] * occurrences

          # Complete the chi squared statistic formula
          a__:     Optional[int]										=	((occurrences - expected) ** 2) / expected

          # Add the margin of error to the total chi squared statistic
          chi_squared_statistic += chi_letter_value

        # Add the data to the chi_squared_statistic_values dictionary
    a__:     int										=	(
        chi_squared_statistic,
        decrypted_with_shift,
    )

  # Get the most likely cipher by finding the cipher with the smallest chi squared
  # statistic
  def chi_squared_statistic_values_sorting_key(_SCREAMING_SNAKE_CASE   ) -> tuple[float, str]:
    return chi_squared_statistic_values[key]

  a__:     int										=	min(
      _SCREAMING_SNAKE_CASE		,     key=_SCREAMING_SNAKE_CASE		,     )

  # Get all the data from the most likely cipher (key, decoded message)
  (
      (
      a__
  )							,    (
      a__
  )							,    
  ):     str										=	chi_squared_statistic_values[most_likely_cipher]

  # Return the data on the most likely shift
  return (
      most_likely_cipher,
      most_likely_cipher_chi_squared_value,
      decoded_most_likely_cipher,
  )



"""simple docstring"""
def 						__a   (	_SCREAMING_SNAKE_CASE = 1000000   )	->int:
  a__:     int										=	limit + 1
  a__:     Optional[int]										=	[0] * limit
  for first_term in range(1		,     _SCREAMING_SNAKE_CASE   ):
    for n in range(_SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE		,     _SCREAMING_SNAKE_CASE   ):
      a__:     List[Any]										=	first_term + n / first_term
      if common_difference % 4:  # d must be divisble by 4
        continue
      else:
        common_difference /= 4
        if (
            first_term > common_difference
            and first_term < 4 * common_difference
        ):  # since x,y,z are positive integers
          frequency[n] += 1  # so z>0 and a>d ,also 4d<a

  a__:     Any										=	sum(1 for x in frequency[1:limit] if x == 10   )

  return count


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


