


from __future__ import annotations

from math import gcd






def 							_lowerCAmelCase (  _a   :					int					,   _a   :					int = 2					,   _a   :					int = 1					,   _a   :					int = 3					,   )	->			int | None:
     # 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(_a   :					int					,   _a   :					int					,   _a   :					int			) -> int:
          return (pow(_a					,   2			) + step) % modulus

     for _ in range(_a			):
          # These track the position within the cycle detection logic.
          lowerCAmelCase_					:       int    =		seed
          lowerCAmelCase_					:       List[str]    =		seed

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

               # 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_					:       Dict    =		gcd(hare - tortoise					,   _a			)

               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_					:       str    =		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_					:   int 							=						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_					:   str 							=						parser.parse_args()

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







from __future__ import annotations






def 							_lowerCAmelCase (  _a   :					list[int]			)	->			list[int]:  # This function is recursive
     lowerCAmelCase_					:       List[Any]    =		len(_a			)
     # If the array contains only one element, we return it (it's the stop condition of
     # recursion)
     if array_length <= 1:
          return array
          # Else
     lowerCAmelCase_					:       Union[str, Any]    =		array[0]
     lowerCAmelCase_					:       str    =		False
     lowerCAmelCase_					:       Optional[int]    =		1
     lowerCAmelCase_					:       list[int]    =		[]
     while not is_found and i < array_length:
          if array[i] < pivot:
               lowerCAmelCase_					:       Optional[Any]    =		True
               lowerCAmelCase_					:       Tuple    =		[element for element in array[i:] if element >= array[i]]
               lowerCAmelCase_					:       Any    =		longest_subsequence(_a			)
               if len(_a			) > len(_a			):
                    lowerCAmelCase_					:       List[Any]    =		temp_array
          else:
               i += 1

     lowerCAmelCase_					:       Tuple    =		[element for element in array[1:] if element >= pivot]
     lowerCAmelCase_					:       List[Any]    =		[pivot, *longest_subsequence(_a			)]
     if len(_a			) > len(_a			):
          return temp_array
     else:
          return longest_subseq


if __name__ == "__main__":
      import doctest

      doctest.testmod()
