





'''simple docstring'''

from __future__ import annotations

from collections.abc import Callable

a        =						list[list[float | int]]

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase      )						->					Matrix:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE      							=       len(__UpperCAmelCase      )
   __SCREAMING_SNAKE_CASE      							=       [[0 for _ in range(size + 1      )] for _ in range(__UpperCAmelCase      )]
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42

   for row in range(__UpperCAmelCase      ):
      for col in range(__UpperCAmelCase      ):
         __SCREAMING_SNAKE_CASE      							=       matrix[row][col]

      __SCREAMING_SNAKE_CASE      							=       vector[row][0]

   __SCREAMING_SNAKE_CASE      							=       0
   __SCREAMING_SNAKE_CASE      							=       0
   while row < size and col < size:
      # pivoting
      __SCREAMING_SNAKE_CASE      							=       max((abs(augmented[rowa][col]      ), rowa) for rowa in range(__UpperCAmelCase							,	__UpperCAmelCase      )      )[
          1
      ]
      if augmented[pivot_row][col] == 0:
         col += 1
         continue
      else:
         __SCREAMING_SNAKE_CASE							,     __SCREAMING_SNAKE_CASE      							=       augmented[pivot_row], augmented[row]

      for rowa in range(row + 1							,	__UpperCAmelCase      ):
         __SCREAMING_SNAKE_CASE      							=       augmented[rowa][col] / augmented[row][col]
         __SCREAMING_SNAKE_CASE      							=       0
         for cola in range(col + 1							,	size + 1      ):
            augmented[rowa][cola] -= augmented[row][cola] * ratio

      row += 1
      col += 1

   # back substitution
   for col in range(1							,	__UpperCAmelCase      ):
      for row in range(__UpperCAmelCase      ):
         __SCREAMING_SNAKE_CASE      							=       augmented[row][col] / augmented[col][col]
         for cola in range(__UpperCAmelCase							,	size + 1      ):
            augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
   return [
       [round(augmented[row][size] / augmented[row][row]							,	10      )] for row in range(__UpperCAmelCase      )
   ]

def 						__magic_name__       (							__UpperCAmelCase      )						->					Callable[[int], int]:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE      							=       len(__UpperCAmelCase      )
   __SCREAMING_SNAKE_CASE      							=       [[0 for _ in range(__UpperCAmelCase      )] for _ in range(__UpperCAmelCase      )]
   __SCREAMING_SNAKE_CASE      							=       [[0] for _ in range(__UpperCAmelCase      )]
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42

   for x_val, y_val in enumerate(__UpperCAmelCase      ):
      for col in range(__UpperCAmelCase      ):
         __SCREAMING_SNAKE_CASE      							=       (x_val + 1) ** (size - col - 1)
      __SCREAMING_SNAKE_CASE      							=       y_val

   __SCREAMING_SNAKE_CASE      							=       solve(__UpperCAmelCase							,	__UpperCAmelCase      )

   def interpolated_func(__UpperCAmelCase      ) -> int:
      return sum(
          round(coeffs[x_val][0]      ) * (var ** (size - x_val - 1))
          for x_val in range(__UpperCAmelCase      )      )

   return interpolated_func

def 						__magic_name__       (							__UpperCAmelCase      )						->					int:

   '''simple docstring'''







   return (
       1
       - variable
       + variable**2
       - variable**3
       + variable**4
       - variable**5
       + variable**6
       - variable**7
       + variable**8
       - variable**9
       + variable**10
   )

def 						__magic_name__       (							__UpperCAmelCase = question_function							,	__UpperCAmelCase = 10      )						->					int:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE      							=       [func(__UpperCAmelCase      ) for x_val in range(1							,	order + 1      )]

   __SCREAMING_SNAKE_CASE      							=       [
       interpolate(data_points[:max_coeff]      ) for max_coeff in range(1							,	order + 1      )
   ]

   __SCREAMING_SNAKE_CASE      							=       0
   __SCREAMING_SNAKE_CASE      							=       42
   __SCREAMING_SNAKE_CASE      							=       42

   for poly in polynomials:
      __SCREAMING_SNAKE_CASE      							=       1
      while func(__UpperCAmelCase      ) == poly(__UpperCAmelCase      ):
         x_val += 1

      ret += poly(__UpperCAmelCase      )

   return ret


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












'''simple docstring'''

import math
from enum import Enum
from typing import Optional, Union

from torch.optim import Optimizer
from torch.optim.lr_scheduler import LambdaLR

from .utils import logging


a        =						logging.get_logger(__name__)







class __a (   _snake_case			):
   __UpperCamelCase   : int	      =	'linear'
   __UpperCamelCase   : Tuple	      =	'cosine'
   __UpperCamelCase   : Tuple	      =	'cosine_with_restarts'
   __UpperCamelCase   : List[Any]	      =	'polynomial'
   __UpperCamelCase   : Optional[Any]	      =	'constant'
   __UpperCamelCase   : Optional[int]	      =	'constant_with_warmup'
   __UpperCamelCase   : List[Any]	      =	'piecewise_constant'





def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase = -1      )						->					int:

   '''simple docstring'''







   return LambdaLR(__UpperCAmelCase							,	lambda __UpperCAmelCase      : 1							,	last_epoch=__UpperCAmelCase      )

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase = -1      )						->					List[Any]:

   '''simple docstring'''







   def lr_lambda(__UpperCAmelCase      ):
      if current_step < num_warmup_steps:
         return float(__UpperCAmelCase      ) / float(max(1.0							,	__UpperCAmelCase      )      )
      return 1.0

   return LambdaLR(__UpperCAmelCase							,	__UpperCAmelCase							,	last_epoch=__UpperCAmelCase      )

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase = -1      )						->					int:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE      							=       {}
   __SCREAMING_SNAKE_CASE      							=       step_rules.split(""","""      )
   for rule_str in rule_list[:-1]:
      __SCREAMING_SNAKE_CASE							,     __SCREAMING_SNAKE_CASE      							=       rule_str.split(""":"""      )
      __SCREAMING_SNAKE_CASE      							=       int(__UpperCAmelCase      )
      __SCREAMING_SNAKE_CASE      							=       float(__UpperCAmelCase      )
      __SCREAMING_SNAKE_CASE      							=       value
   __SCREAMING_SNAKE_CASE      							=       float(rule_list[-1]      )

   def create_rules_function(__UpperCAmelCase							,	__UpperCAmelCase      ):
      def rule_func(__UpperCAmelCase      ) -> float:
         __SCREAMING_SNAKE_CASE      							=       sorted(rules_dict.keys()      )
         for i, sorted_step in enumerate(__UpperCAmelCase      ):
            if steps < sorted_step:
               return rules_dict[sorted_steps[i]]
         return last_lr_multiple

      return rule_func

   __SCREAMING_SNAKE_CASE      							=       create_rules_function(__UpperCAmelCase							,	__UpperCAmelCase      )

   return LambdaLR(__UpperCAmelCase							,	__UpperCAmelCase							,	last_epoch=__UpperCAmelCase      )

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase=-1      )						->					int:

   '''simple docstring'''







   def lr_lambda(__UpperCAmelCase      ):
      if current_step < num_warmup_steps:
         return float(__UpperCAmelCase      ) / float(max(1							,	__UpperCAmelCase      )      )
      return max(
          0.0							,	float(num_training_steps - current_step      ) / float(max(1							,	num_training_steps - num_warmup_steps      )      )      )

   return LambdaLR(__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase      )

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase = 0.5							,	__UpperCAmelCase = -1      )						->					Dict:

   '''simple docstring'''







   def lr_lambda(__UpperCAmelCase      ):
      if current_step < num_warmup_steps:
         return float(__UpperCAmelCase      ) / float(max(1							,	__UpperCAmelCase      )      )
      __SCREAMING_SNAKE_CASE      							=       float(current_step - num_warmup_steps      ) / float(max(1							,	num_training_steps - num_warmup_steps      )      )
      return max(0.0							,	0.5 * (1.0 + math.cos(math.pi * float(__UpperCAmelCase      ) * 2.0 * progress      ))      )

   return LambdaLR(__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase      )

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase = 1							,	__UpperCAmelCase = -1      )						->					Tuple:

   '''simple docstring'''







   def lr_lambda(__UpperCAmelCase      ):
      if current_step < num_warmup_steps:
         return float(__UpperCAmelCase      ) / float(max(1							,	__UpperCAmelCase      )      )
      __SCREAMING_SNAKE_CASE      							=       float(current_step - num_warmup_steps      ) / float(max(1							,	num_training_steps - num_warmup_steps      )      )
      if progress >= 1.0:
         return 0.0
      return max(0.0							,	0.5 * (1.0 + math.cos(math.pi * ((float(__UpperCAmelCase      ) * progress) % 1.0)      ))      )

   return LambdaLR(__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase      )

def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase=1e-7							,	__UpperCAmelCase=1.0							,	__UpperCAmelCase=-1      )						->					Tuple:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE      							=       optimizer.defaults["""lr"""]
   if not (lr_init > lr_end):
      raise ValueError(f"""lr_end ({lr_end}) must be be smaller than initial lr ({lr_init})"""      )

   def lr_lambda(__UpperCAmelCase      ):
      if current_step < num_warmup_steps:
         return float(__UpperCAmelCase      ) / float(max(1							,	__UpperCAmelCase      )      )
      elif current_step > num_training_steps:
         return lr_end / lr_init  # as LambdaLR multiplies by lr_init
      else:
         __SCREAMING_SNAKE_CASE      							=       lr_init - lr_end
         __SCREAMING_SNAKE_CASE      							=       num_training_steps - num_warmup_steps
         __SCREAMING_SNAKE_CASE      							=       1 - (current_step - num_warmup_steps) / decay_steps
         __SCREAMING_SNAKE_CASE      							=       lr_range * pct_remaining**power + lr_end
         return decay / lr_init  # as LambdaLR multiplies by lr_init

   return LambdaLR(__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase      )


a        =						{
    SchedulerType.LINEAR: get_linear_schedule_with_warmup,
    SchedulerType.COSINE: get_cosine_schedule_with_warmup,
    SchedulerType.COSINE_WITH_RESTARTS: get_cosine_with_hard_restarts_schedule_with_warmup,
    SchedulerType.POLYNOMIAL: get_polynomial_decay_schedule_with_warmup,
    SchedulerType.CONSTANT: get_constant_schedule,
    SchedulerType.CONSTANT_WITH_WARMUP: get_constant_schedule_with_warmup,
    SchedulerType.PIECEWISE_CONSTANT: get_piecewise_constant_schedule,
}







def 						__magic_name__       (							__UpperCAmelCase							,	__UpperCAmelCase							,	__UpperCAmelCase = None							,	__UpperCAmelCase = None							,	__UpperCAmelCase = None							,	__UpperCAmelCase = 1							,	__UpperCAmelCase = 1.0							,	__UpperCAmelCase = -1							,	)						->					str:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE      							=       SchedulerType(__UpperCAmelCase      )
   __SCREAMING_SNAKE_CASE      							=       TYPE_TO_SCHEDULER_FUNCTION[name]
   if name == SchedulerType.CONSTANT:
      return schedule_func(__UpperCAmelCase							,	last_epoch=__UpperCAmelCase      )

   if name == SchedulerType.PIECEWISE_CONSTANT:
      return schedule_func(__UpperCAmelCase							,	step_rules=__UpperCAmelCase							,	last_epoch=__UpperCAmelCase      )

   # All other schedulers require `num_warmup_steps`
   if num_warmup_steps is None:
      raise ValueError(f"""{name} requires `num_warmup_steps`, please provide that argument."""      )

   if name == SchedulerType.CONSTANT_WITH_WARMUP:
      return schedule_func(__UpperCAmelCase							,	num_warmup_steps=__UpperCAmelCase							,	last_epoch=__UpperCAmelCase      )

   # All other schedulers require `num_training_steps`
   if num_training_steps is None:
      raise ValueError(f"""{name} requires `num_training_steps`, please provide that argument."""      )

   if name == SchedulerType.COSINE_WITH_RESTARTS:
      return schedule_func(
          __UpperCAmelCase							,	num_warmup_steps=__UpperCAmelCase							,	num_training_steps=__UpperCAmelCase							,	num_cycles=__UpperCAmelCase							,	last_epoch=__UpperCAmelCase							,	)

   if name == SchedulerType.POLYNOMIAL:
      return schedule_func(
          __UpperCAmelCase							,	num_warmup_steps=__UpperCAmelCase							,	num_training_steps=__UpperCAmelCase							,	power=__UpperCAmelCase							,	last_epoch=__UpperCAmelCase							,	)

   return schedule_func(
       __UpperCAmelCase							,	num_warmup_steps=__UpperCAmelCase							,	num_training_steps=__UpperCAmelCase							,	last_epoch=__UpperCAmelCase      )





