


from functools import reduce

a_ :Optional[Any]  							=			(
    '73167176531330624919225119674426574742355349194934'
    '96983520312774506326239578318016984801869478851843'
    '85861560789112949495459501737958331952853208805511'
    '12540698747158523863050715693290963295227443043557'
    '66896648950445244523161731856403098711121722383113'
    '62229893423380308135336276614282806444486645238749'
    '30358907296290491560440772390713810515859307960866'
    '70172427121883998797908792274921901699720888093776'
    '65727333001053367881220235421809751254540594752243'
    '52584907711670556013604839586446706324415722155397'
    '53697817977846174064955149290862569321978468622482'
    '83972241375657056057490261407972968652414535100474'
    '82166370484403199890008895243450658541227588666881'
    '16427171479924442928230863465674813919123162824586'
    '17866458359124566529476545682848912883142607690042'
    '24219022671055626321111109370544217506941658960408'
    '07198403850962455444362981230987879927244284909188'
    '84580156166097919133875499200524063689912560717606'
    '05886116467109405077541002256983155200055935729725'
    '71636269561882670428252483600823257530420752963450'
)




def    a   (							A__ = N							)	-> int:




      '''simple docstring'''

      return max(
          # mypy cannot properly interpret reduce
          int(reduce(lambda A__      ,    A__							: str(int(A__							) * int(A__							)							)      ,    n[i : i + 1_3]							)							)
          for i in range(len(A__							) - 1_2							)							)


if __name__ == "__main__":
    print(F'''{solution() = }''')








from dataclasses import dataclass
from typing import Optional, Tuple, Union

import flax
import jax.numpy as jnp
from jax import random

from ..configuration_utils import ConfigMixin, register_to_config
from ..utils import BaseOutput
from .scheduling_utils_flax import FlaxSchedulerMixin
@flax.struct.dataclass
class       _lowerCAmelCase  :

 """simple docstring"""
 # setable values
 SCREAMING_SNAKE_CASE_				:					Optional[int]       						=None
 SCREAMING_SNAKE_CASE_				:					Optional[jnp.ndarray]       						=None
 SCREAMING_SNAKE_CASE_				:					Optional[jnp.ndarray]       						=None  # sigma(t_i)

 @classmethod
 def      __lowerCAmelCase  (						cls			:			Optional[Any]   ):






      """simple docstring"""



      return cls()




@dataclass
class       _lowerCAmelCase  (		__magic_name__							):

 """simple docstring"""
 SCREAMING_SNAKE_CASE_				:					jnp.ndarray
 SCREAMING_SNAKE_CASE_				:					jnp.ndarray
 SCREAMING_SNAKE_CASE_				:					KarrasVeSchedulerState




class       _lowerCAmelCase  (		__magic_name__			,	__magic_name__							):

 """simple docstring"""

 @property
 def      __lowerCAmelCase  (						self			:			Any   ):






      """simple docstring"""



      return True

 @register_to_config
 def __init__(						self			:			Dict					, SCREAMING_SNAKE_CASE__			:			float = 0.02					, SCREAMING_SNAKE_CASE__			:			float = 1_00					, SCREAMING_SNAKE_CASE__			:			float = 1.007					, SCREAMING_SNAKE_CASE__			:			float = 80					, SCREAMING_SNAKE_CASE__			:			float = 0.05					, SCREAMING_SNAKE_CASE__			:			float = 50					, ):






      """simple docstring"""



      pass

 def      __lowerCAmelCase  (						self			:			int   ):






      """simple docstring"""



      return KarrasVeSchedulerState.create()

 def      __lowerCAmelCase  (						self			:			Union[str, Any]					, SCREAMING_SNAKE_CASE__			:			KarrasVeSchedulerState					, SCREAMING_SNAKE_CASE__			:			int					, SCREAMING_SNAKE_CASE__			:			Tuple = ()   ):






      """simple docstring"""



      UpperCamelCase						    =					jnp.arange(0					, SCREAMING_SNAKE_CASE__   )[::-1].copy()
      UpperCamelCase						    =					[
          (
              self.config.sigma_max**2
              * (self.config.sigma_min**2 / self.config.sigma_max**2) ** (i / (num_inference_steps - 1))
          )
          for i in timesteps
      ]

      return state.replace(
          num_inference_steps=SCREAMING_SNAKE_CASE__					, schedule=jnp.array(SCREAMING_SNAKE_CASE__					, dtype=jnp.floataa   )					, timesteps=SCREAMING_SNAKE_CASE__					, )

 def      __lowerCAmelCase  (						self			:			List[Any]					, SCREAMING_SNAKE_CASE__			:			KarrasVeSchedulerState					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			float					, SCREAMING_SNAKE_CASE__			:			random.KeyArray					, ):






      """simple docstring"""



      if self.config.s_min <= sigma <= self.config.s_max:
           UpperCamelCase						    =					min(self.config.s_churn / state.num_inference_steps					, 2**0.5 - 1   )
      else:
           UpperCamelCase						    =					0

      # sample eps ~ N(0, S_noise^2 * I)
      UpperCamelCase						    =					random.split(SCREAMING_SNAKE_CASE__					, num=1   )
      UpperCamelCase						    =					self.config.s_noise * random.normal(key=SCREAMING_SNAKE_CASE__					, shape=sample.shape   )
      UpperCamelCase						    =					sigma + gamma * sigma
      UpperCamelCase						    =					sample + ((sigma_hat**2 - sigma**2) ** 0.5 * eps)

      return sample_hat, sigma_hat

 def      __lowerCAmelCase  (						self			:			Optional[int]					, SCREAMING_SNAKE_CASE__			:			KarrasVeSchedulerState					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			float					, SCREAMING_SNAKE_CASE__			:			float					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			bool = True					, ):






      """simple docstring"""



      UpperCamelCase						    =					sample_hat + sigma_hat * model_output
      UpperCamelCase						    =					(sample_hat - pred_original_sample) / sigma_hat
      UpperCamelCase						    =					sample_hat + (sigma_prev - sigma_hat) * derivative

      if not return_dict:
           return (sample_prev, derivative, state)

      return FlaxKarrasVeOutput(prev_sample=SCREAMING_SNAKE_CASE__					, derivative=SCREAMING_SNAKE_CASE__					, state=SCREAMING_SNAKE_CASE__   )

 def      __lowerCAmelCase  (						self			:			Tuple					, SCREAMING_SNAKE_CASE__			:			KarrasVeSchedulerState					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			float					, SCREAMING_SNAKE_CASE__			:			float					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			jnp.ndarray					, SCREAMING_SNAKE_CASE__			:			bool = True					, ):






      """simple docstring"""



      UpperCamelCase						    =					sample_prev + sigma_prev * model_output
      UpperCamelCase						    =					(sample_prev - pred_original_sample) / sigma_prev
      UpperCamelCase						    =					sample_hat + (sigma_prev - sigma_hat) * (0.5 * derivative + 0.5 * derivative_corr)

      if not return_dict:
           return (sample_prev, derivative, state)

      return FlaxKarrasVeOutput(prev_sample=SCREAMING_SNAKE_CASE__					, derivative=SCREAMING_SNAKE_CASE__					, state=SCREAMING_SNAKE_CASE__   )






 def      __lowerCAmelCase  (						self			:			Any					, SCREAMING_SNAKE_CASE__			:			KarrasVeSchedulerState					, SCREAMING_SNAKE_CASE__			:			Any					, SCREAMING_SNAKE_CASE__			:			str					, SCREAMING_SNAKE_CASE__			:			int   ):






      """simple docstring"""



      raise NotImplementedError()


