
import math

def 		UpperCamelCase	(      lowercase_   ):


  '''simple docstring'''






  return math.sqrt(_UpperCamelCase   ) * math.sqrt(_UpperCamelCase   ) == num

def 		UpperCamelCase	(      lowercase_   ):


  '''simple docstring'''






  lowercase__  :   Union[str, Any]    		=      0
  lowercase__  :   Optional[int]    		=      n
  while left <= right:
    lowercase__  :   int    		=      (left + right) // 2
    if mid**2 == n:
      return True
    elif mid**2 > n:
      lowercase__  :   List[str]    		=      mid - 1
    else:
      lowercase__  :   List[Any]    		=      mid + 1
  return False


if __name__ == "__main__":
     import doctest

     doctest.testmod()








from math import factorial

lowerCamelCase__    :		dict[str, int]          =   {str(digit): factorial(digit) for digit in range(1_0)}

def 		UpperCamelCase	(      lowercase_   )		->	int:


  '''simple docstring'''






  if not isinstance(lowercase_ , lowercase_   ):
    raise TypeError("""Parameter number must be int"""   )

  if number < 0:
    raise ValueError("""Parameter number must be greater than or equal to 0"""   )

  # Converts number in string to iterate on its digits and adds its factorial.
  return sum(DIGIT_FACTORIAL[digit] for digit in str(lowercase_   )   )

def 		UpperCamelCase	(      lowercase_ = 60 , lowercase_ = 1_00_00_00   )		->	int:


  '''simple docstring'''






  if not isinstance(lowercase_ , lowercase_   ) or not isinstance(lowercase_ , lowercase_   ):
    raise TypeError("""Parameters chain_length and number_limit must be int"""   )

  if chain_length <= 0 or number_limit <= 0:
    raise ValueError(
        """Parameters chain_length and number_limit must be greater than 0"""   )

  # the counter for the chains with the exact desired length
  lowercase__  :   List[Any]    		=      0
  # the cached sizes of the previous chains
  lowercase__  :   dict[int, int]    		=      {}

  for start_chain_element in range(1 , lowercase_   ):
    # The temporary set will contain the elements of the chain
    lowercase__  :   Optional[Any]    		=      set()
    lowercase__  :   Union[str, Any]    		=      0

    # Stop computing the chain when you find a cached size, a repeating item or the
    # length is greater then the desired one.
    lowercase__  :   Union[str, Any]    		=      start_chain_element
    while (
        chain_element not in chain_sets_lengths
        and chain_element not in chain_set
        and chain_set_length <= chain_length
    ):
      chain_set.add(lowercase_   )
      chain_set_length += 1
      lowercase__  :   str    		=      digit_factorial_sum(lowercase_   )

    if chain_element in chain_sets_lengths:
      chain_set_length += chain_sets_lengths[chain_element]

    lowercase__  :   str    		=      chain_set_length

    # If chain contains the exact amount of elements increase the counter
    if chain_set_length == chain_length:
      chains_counter += 1

  return chains_counter


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     print(f'''{solution()}''')






