

from __future__ import annotations

from math import gcd







def       lowercase					(	__A :					int ,    __A :					int = 2 ,    __A :					int = 1 ,    __A :					int = 3 ,    )				->			int | None:





    '''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 :					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.
        snake_case							:		str        =		seed
        snake_case							:		int        =		seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            snake_case							:		Union[str, Any]        =		rand_fn(__A ,    __A ,    __A		)
            snake_case							:		Dict        =		rand_fn(__A ,    __A ,    __A		)
            snake_case							:		int        =		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``.
            snake_case							:		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.
        snake_case							:		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

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

     __lowercase     : Tuple       =						pollard_rho(args.num, attempts=args.attempts)
     if divisor is None:
          print(f'''{args.num} is probably prime''')
     else:
          __lowercase     : Optional[int]       =						args.num // divisor
          print(f'''{args.num} = {divisor} * {quotient}''')







'''simple docstring'''

from __future__ import annotations

from collections import namedtuple
from dataclasses import dataclass
@dataclass
class     __lowerCAmelCase		:





      '''simple docstring'''






      lowerCAmelCase__   :	int
      lowerCAmelCase__   :	TreeNode | None       =				None
      lowerCAmelCase__   :	TreeNode | None       =				None






lowerCamelCase				:  Optional[Any]    		=      namedtuple('CoinsDistribResult', 'moves excess')

def _SCREAMING_SNAKE_CASE (A			)			->			int:






       """simple docstring"""







       if root is None:
              return 0

       # Validation
       def count_nodes(A			) -> int:
              if node is None:
                     return 0

              return count_nodes(node.left			) + count_nodes(node.right			) + 1

       def count_coins(A			) -> int:
              if node is None:
                     return 0

              return count_coins(node.left			) + count_coins(node.right			) + node.data

       if count_nodes(A			) != count_coins(A			):
              raise ValueError('''The nodes number should be same as the number of coins'''			)

       # Main calculation
       def get_distrib(A			) -> CoinsDistribResult:

              if node is None:
                     return CoinsDistribResult(0    ,					1			)

              lowercase__     ,lowercase__      =					get_distrib(node.left			)
              lowercase__     ,lowercase__      =					get_distrib(node.right			)

              lowercase__      =					1 - left_distrib_excess
              lowercase__      =					1 - right_distrib_excess

              lowercase__      =					(
                  left_distrib_moves
                  + right_distrib_moves
                  + abs(A			)
                  + abs(A			)
              )
              lowercase__      =					node.data - coins_to_left - coins_to_right

              return CoinsDistribResult(A    ,					A			)

       return get_distrib(A			)[0]


if __name__ == "__main__":
 import doctest

 doctest.testmod()




