from __future__ import annotations

from math import gcd





def    __lowercase     (						lowerCamelCase					:     int  ,  lowerCamelCase					:     int = 2  ,  lowerCamelCase					:     int = 1  ,  lowerCamelCase					:     int = 3  ,  ):
 # 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.
  UpperCamelCase_					:						Any =					seed
  UpperCamelCase_					:						int =					seed

  while True:
   # At each iteration, the tortoise moves one step and the hare moves two.
   UpperCamelCase_					:						List[Any] =					rand_fn(lowerCamelCase  ,  lowerCamelCase  ,  lowerCamelCase	)
   UpperCamelCase_					:						Optional[Any] =					rand_fn(lowerCamelCase  ,  lowerCamelCase  ,  lowerCamelCase	)
   UpperCamelCase_					:						Optional[int] =					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``.
   UpperCamelCase_					:						int =					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.
  UpperCamelCase_					:						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_    					=       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_    					=       parser.parse_args()

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




from typing import TYPE_CHECKING

from ...utils import (
    OptionalDependencyNotAvailable,
    _LazyModule,
    is_flax_available,
    is_tf_available,
    is_torch_available,
)


a_    					=       {'configuration_encoder_decoder': ['EncoderDecoderConfig']}

try:
   if not is_torch_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   a_    					=       ['EncoderDecoderModel']

try:
   if not is_tf_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   a_    					=       ['TFEncoderDecoderModel']

try:
   if not is_flax_available():
      raise OptionalDependencyNotAvailable()
except OptionalDependencyNotAvailable:
   pass
else:
   a_    					=       ['FlaxEncoderDecoderModel']

if TYPE_CHECKING:
   from .configuration_encoder_decoder import EncoderDecoderConfig

   try:
      if not is_torch_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .modeling_encoder_decoder import EncoderDecoderModel

   try:
      if not is_tf_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .modeling_tf_encoder_decoder import TFEncoderDecoderModel

   try:
      if not is_flax_available():
         raise OptionalDependencyNotAvailable()
   except OptionalDependencyNotAvailable:
      pass
   else:
      from .modeling_flax_encoder_decoder import FlaxEncoderDecoderModel

else:
   import sys

   a_    					=       _LazyModule(__name__, globals()['__file__'], _import_structure, module_spec=__spec__)



