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.
        lowercase__ :   Optional[int]    					=				seed
        lowercase__ :   int    					=				seed

        while True:
            # At each iteration, the tortoise moves one step and the hare moves two.
            lowercase__ :   List[Any]    					=				rand_fn(_lowerCamelCase		,	_lowerCamelCase		,	_lowerCamelCase)
            lowercase__ :   List[str]    					=				rand_fn(_lowerCamelCase		,	_lowerCamelCase		,	_lowerCamelCase)
            lowercase__ :   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``.
            lowercase__ :   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.
        lowercase__ :   Dict    					=				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  							=						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  							=						parser.parse_args()

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











from typing import Mapping

from ...configuration_utils import PretrainedConfig
from ...onnx import OnnxSeqaSeqConfigWithPast
from ...utils import logging


UpperCAmelCase__				   =		logging.get_logger(__name__)

UpperCAmelCase__				   =		{
    "t5-small": "https://huggingface.co/t5-small/resolve/main/config.json",
    "t5-base": "https://huggingface.co/t5-base/resolve/main/config.json",
    "t5-large": "https://huggingface.co/t5-large/resolve/main/config.json",
    "t5-3b": "https://huggingface.co/t5-3b/resolve/main/config.json",
    "t5-11b": "https://huggingface.co/t5-11b/resolve/main/config.json",
}






class 		lowercase_					(	lowercase   ):






      '''simple docstring'''




      __snake_case       =  '''t5'''
      __snake_case       =  ['''past_key_values''']
      __snake_case       =  {'''hidden_size''': '''d_model''', '''num_attention_heads''': '''num_heads''', '''num_hidden_layers''': '''num_layers'''}







      def __init__(      self      :						Optional[Any] ,					__UpperCAmelCase      :						Optional[Any]=32_128 ,					__UpperCAmelCase      :						List[Any]=512 ,					__UpperCAmelCase      :						Dict=64 ,					__UpperCAmelCase      :						Tuple=2_048 ,					__UpperCAmelCase      :						int=6 ,					__UpperCAmelCase      :						Optional[int]=None ,					__UpperCAmelCase      :						Optional[int]=8 ,					__UpperCAmelCase      :						str=32 ,					__UpperCAmelCase      :						Tuple=128 ,					__UpperCAmelCase      :						Optional[Any]=0.1 ,					__UpperCAmelCase      :						int=1e-6 ,					__UpperCAmelCase      :						int=1.0 ,					__UpperCAmelCase      :						List[str]="relu" ,					__UpperCAmelCase      :						int=True ,					__UpperCAmelCase      :						int=True ,					__UpperCAmelCase      :						List[Any]=0 ,					__UpperCAmelCase      :						int=1 ,					**__UpperCAmelCase      :						str ,					)    ->Optional[Any]:




       """simple docstring"""


       a		  =      vocab_size
       a		  =      d_model
       a		  =      d_kv
       a		  =      d_ff
       a		  =      num_layers
       a		  =      (
           num_decoder_layers if num_decoder_layers is not None else self.num_layers
       )  # default = symmetry
       a		  =      num_heads
       a		  =      relative_attention_num_buckets
       a		  =      relative_attention_max_distance
       a		  =      dropout_rate
       a		  =      layer_norm_epsilon
       a		  =      initializer_factor
       a		  =      feed_forward_proj
       a		  =      use_cache

       a		  =      self.feed_forward_proj.split('''-'''  )
       a		  =      act_info[-1]
       a		  =      act_info[0] == '''gated'''

       if len(__UpperCAmelCase  ) > 1 and act_info[0] != "gated" or len(__UpperCAmelCase  ) > 2:
        raise ValueError(
            F"""`feed_forward_proj`: {feed_forward_proj} is not a valid activation function of the dense layer."""
            '''Please make sure `feed_forward_proj` is of the format `gated-{ACT_FN}` or `{ACT_FN}`, e.g. '''
            '''\'gated-gelu\' or \'relu\''''  )

       # for backwards compatibility
       if feed_forward_proj == "gated-gelu":
        a		  =      '''gelu_new'''

       super().__init__(
           pad_token_id=__UpperCAmelCase ,					eos_token_id=__UpperCAmelCase ,					is_encoder_decoder=__UpperCAmelCase ,					**__UpperCAmelCase ,					)






class 		lowercase_					(	lowercase   ):






      '''simple docstring'''




      @property
      def       __lowerCAmelCase					(      self      :						Optional[Any]  )    ->Mapping[str, Mapping[int, str]]:




       """simple docstring"""


       a		  =      {
           '''input_ids''': {0: '''batch''', 1: '''encoder_sequence'''},
           '''attention_mask''': {0: '''batch''', 1: '''encoder_sequence'''},
       }
       if self.use_past:
        a		  =      '''past_encoder_sequence + sequence'''
        a		  =      {0: '''batch'''}
        a		  =      {0: '''batch''', 1: '''past_decoder_sequence + sequence'''}
       else:
        a		  =      {0: '''batch''', 1: '''decoder_sequence'''}
        a		  =      {0: '''batch''', 1: '''decoder_sequence'''}

       if self.use_past:
        self.fill_with_past_key_values_(__UpperCAmelCase ,					direction='''inputs'''  )

       return common_inputs







      @property
      def       __lowerCAmelCase					(      self      :						Union[str, Any]  )    ->int:




       """simple docstring"""


       return 13


