


'''simple docstring'''







from __future__ import annotations

from collections.abc import Iterator
class 						UpperCamelCase__       :



     """simple docstring"""






     def __init__(			self   ,						snake_case__ ):

            '''simple docstring'''

            _lowerCAmelCase     :				List[Any]       =							value
            _lowerCAmelCase     :				Node | None       =							None
            _lowerCAmelCase     :				Node | None       =							None
class 						UpperCamelCase__       :



     """simple docstring"""






     def __init__(			self   ,						snake_case__ ):

            '''simple docstring'''

            _lowerCAmelCase     :				List[Any]       =							tree







     def a						(			self   ,						snake_case__ ):

            '''simple docstring'''

            if node is None:
                   return 0
            return node.value + (
                self.depth_first_search(node.left ) + self.depth_first_search(node.right )
            )







     def __iter__(			self ):

            '''simple docstring'''

            yield self.depth_first_search(self.tree )


if __name__ == "__main__":
       import doctest

       doctest.testmod()





'''simple docstring'''







from __future__ import annotations

from math import gcd






def       lowercase					(_A ,    _A = 2 ,    _A = 1 ,    _A = 3 ,    ):







       """simple docstring"""







       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 ,    _A ,    _A       ) -> int:
              return (pow(_A ,    2       ) + step) % modulus

       for _ in range(_A       ):
              # These track the position within the cycle detection logic.
              _lowerCAmelCase     :				Dict       =							seed
              _lowerCAmelCase     :				int       =							seed

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

       lowerCAmelCase     :  Tuple											=    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""",
       )
       lowerCAmelCase     :  List[str]											=    parser.parse_args()

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

