"""simple docstring"""




from __future__ import annotations

from math import gcd





def 							SCREAMING_SNAKE_CASE     (			_lowerCamelCase						:       int			,_lowerCamelCase						:       int = 2			,_lowerCamelCase						:       int = 1			,_lowerCamelCase						:       int = 3			,)       ->      Union[str, Any]:
  # A value less than 2 can cause an infinite loop in the algorithm.
  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(_lowerCamelCase						:       int			,_lowerCamelCase						:       int			,_lowerCamelCase						:       int	) -> int:
    return (pow(_lowerCamelCase			,2	) + step) % modulus

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

    while True:
      # At each iteration, the tortoise moves one step and the hare moves two.
      _lowerCAmelCase						:       Optional[Any]  			=     rand_fn(_lowerCamelCase			,_lowerCamelCase			,_lowerCamelCase	)
      _lowerCAmelCase						:       List[Any]  			=     rand_fn(_lowerCamelCase			,_lowerCamelCase			,_lowerCamelCase	)
      _lowerCAmelCase						:       Union[str, Any]  			=     rand_fn(_lowerCamelCase			,_lowerCamelCase			,_lowerCamelCase	)

      # 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``.
      _lowerCAmelCase						:       Union[str, Any]  			=     gcd(hare - tortoise			,_lowerCamelCase	)

      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.
    _lowerCAmelCase						:       List[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

     _a      :		List[Any]           =   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',
     )
     _a      :		str           =   parser.parse_args()

     _a      :		Any           =   pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(F"""{args.num} is probably prime""")
     else:
          _a      :		List[str]           =   args.num // divisor
          print(F"""{args.num} = {divisor} * {quotient}""")











'''simple docstring'''

from __future__ import annotations





def 						UpperCamelCase				(							_lowerCamelCase					:					list[int | str]       ):
    create_state_space_tree(_lowerCamelCase						,       []						,       0						,       [0 for i in range(len(_lowerCamelCase       )       )]       )





def 						UpperCamelCase				(							_lowerCamelCase					:					list[int | str]						,       _lowerCamelCase					:					list[int | str]						,       _lowerCamelCase					:					int						,       _lowerCamelCase					:					list[int]						,       ):

    if index == len(_lowerCamelCase       ):
        print(_lowerCamelCase       )
        return

    for i in range(len(_lowerCamelCase       )       ):
        if not index_used[i]:
            current_sequence.append(sequence[i]       )
            A__				=			True
            create_state_space_tree(_lowerCamelCase						,       _lowerCamelCase						,       index + 1						,       _lowerCamelCase       )
            current_sequence.pop()
            A__				=			False
__lowerCAmelCase							:	list[int | str]			=[3, 1, 2, 4]
generate_all_permutations(sequence)

__lowerCAmelCase							:	list[int | str]			=["A", "B", "C"]



generate_all_permutations(sequence_a)



