


from __future__ import annotations

from math import gcd


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE = 2 ,			SCREAMING_SNAKE_CASE = 1 ,			SCREAMING_SNAKE_CASE = 3 ,			):

  if num < 2:
    raise ValueError('''The input value cannot be less than 2'''       )

  # Because of the relationship between ``f(f(x))`` and ``f(x)``, this
  # algorithm struggles to find factors that are divisible by two.
  # As a workaround, we specifically check for two and even inputs.
  #   See: https://math.stackexchange.com/a/2856214/165820
  if num > 2 and num % 2 == 0:
    return 2

  # Pollard's Rho algorithm requires a function that returns pseudorandom
  # values between 0 <= X < ``num``.  It doesn't need to be random in the
  # sense that the output value is cryptographically secure or difficult
  # to calculate, it only needs to be random in the sense that all output
  # values should be equally likely to appear.
  # For this reason, Pollard suggested using ``f(x) = (x**2 - 1) % num``
  # However, the success of Pollard's algorithm isn't guaranteed and is
  # determined in part by the initial seed and the chosen random function.
  # To make retries easier, we will instead use ``f(x) = (x**2 + C) % num``
  # where ``C`` is a value that we can modify between each attempt.
  def rand_fn(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ) -> int:
    return (pow(SCREAMING_SNAKE_CASE ,			2       ) + step) % modulus

  for _ in range(SCREAMING_SNAKE_CASE       ):
    # These track the position within the cycle detection logic.
    A_      :				Union[str, Any]		     =					seed
    A_      :				str		     =					seed

    while True:
      # At each iteration, the tortoise moves one step and the hare moves two.
      A_      :				int		     =					rand_fn(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       )
      A_      :				Optional[int]		     =					rand_fn(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       )
      A_      :				List[str]		     =					rand_fn(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       )

      # At some point both the tortoise and the hare will enter a cycle whose
      # length ``p`` is a divisor of ``num``.  Once in that cycle, at some point
      # the tortoise and hare will end up on the same value modulo ``p``.
      # We can detect when this happens because the position difference between
      # the tortoise and the hare will share a common divisor with ``num``.
      A_      :				Any		     =					gcd(hare - tortoise ,			SCREAMING_SNAKE_CASE       )

      if divisor == 1:
        # No common divisor yet, just keep searching.
        continue
      else:
        # We found a common divisor!
        if divisor == num:
          # Unfortunately, the divisor is ``num`` itself and is useless.
          break
        else:
          # The divisor is a nontrivial factor of ``num``!
          return divisor

        # If we made it here, then this attempt failed.
        # We need to pick a new starting seed for the tortoise and hare
        # in addition to a new step value for the random function.
        # To keep this example implementation deterministic, the
        # new values will be generated based on currently available
        # values instead of using something like ``random.randint``.

        # We can use the hare's position as the new seed.
        # This is actually what Richard Brent's the "optimized" variant does.
    A_      :				Optional[Any]		     =					hare

    # The new step value for the random function can just be incremented.
    # At first the results will be similar to what the old function would
    # have produced, but the value will quickly diverge after a bit.
    step += 1

  # We haven't found a divisor within the requested number of attempts.
  # We were unlucky or ``num`` itself is actually prime.
  return None


if __name__ == "__main__":
     import argparse

     UpperCamelCase								=	argparse.ArgumentParser()
     parser.add_argument(
         """num""",
         type=int,
         help="""The value to find a divisor of""",
     )
     parser.add_argument(
         """--attempts""",
         type=int,
         default=3,
         help="""The number of attempts before giving up""",
     )
     UpperCamelCase								=	parser.parse_args()

     UpperCamelCase								=	pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(F'''{args.num} is probably prime''')
     else:
          UpperCamelCase								=	args.num // divisor
          print(F'''{args.num} = {divisor} * {quotient}''')






def       A       (					lowercase )   ->  list:
		'''simple docstring'''

		UpperCamelCase		=     len(lowercase )
		for i in range(1    ,       lowercase ):
				UpperCamelCase		=     collection[i]
				UpperCamelCase		=     0
				UpperCamelCase		=     i - 1

				while low <= high:
						UpperCamelCase		=     (low + high) // 2
						if val < collection[mid]:
								UpperCamelCase		=     mid - 1
						else:
								UpperCamelCase		=     mid + 1
				for j in range(lowercase    ,       lowercase    ,       -1 ):
						UpperCamelCase		=     collection[j - 1]
				UpperCamelCase		=     val
		return collection


if __name__ == "__main__":
					_UpperCAmelCase					: List[Any]      =       input("Enter numbers separated by a comma:\n").strip()
					_UpperCAmelCase					: Optional[int]      =       [int(item) for item in user_input.split(",")]
					print(binary_insertion_sort(unsorted))
