
from math import pow






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_	,	UpperCamelCase_	,	)		->		tuple[int, int]:
					'''simple docstring'''


					if current_sum == needed_sum:
										# If the sum of the powers is equal to needed_sum, then we have a solution.
										solutions_count += 1
										return current_sum, solutions_count

					SCREAMING_SNAKE_CASE__													= int(pow(UpperCamelCase_	,	UpperCamelCase_		)		)
					if current_sum + i_to_n <= needed_sum:
										# If the sum of the powers is less than needed_sum, then continue adding powers.
										current_sum += i_to_n
										SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= backtrack(
										    UpperCamelCase_	,	UpperCamelCase_	,	current_number + 1	,	UpperCamelCase_	,	UpperCamelCase_		)
										current_sum -= i_to_n
					if i_to_n < needed_sum:
										# If the power of i is less than needed_sum, then try with the next power.
										SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= backtrack(
										    UpperCamelCase_	,	UpperCamelCase_	,	current_number + 1	,	UpperCamelCase_	,	UpperCamelCase_		)
					return current_sum, solutions_count






def 							_lowercase							(     UpperCamelCase_	,	UpperCamelCase_		)		->		int:
					'''simple docstring'''


					if not (1 <= needed_sum <= 1000 and 2 <= power <= 10):
										raise ValueError(
										    'Invalid input\n'
										    'needed_sum must be between 1 and 1000, power between 2 and 10.'		)

					return backtrack(UpperCamelCase_	,	UpperCamelCase_	,	1	,	0	,	0		)[1]  # Return the solutions_count


if __name__ == "__main__":
				import doctest

				doctest.testmod()







from typing import Any, Dict, List, Optional, Tuple, Union

import torch
from torch import nn
from torch.utils.data import DistributedSampler, RandomSampler

from transformers import PreTrainedModel, Trainer, logging
from transformers.integrations import is_fairscale_available
from transformers.models.fsmt.configuration_fsmt import FSMTConfig
from transformers.optimization import (
    Adafactor,
    AdamW,
    get_constant_schedule,
    get_constant_schedule_with_warmup,
    get_cosine_schedule_with_warmup,
    get_cosine_with_hard_restarts_schedule_with_warmup,
    get_linear_schedule_with_warmup,
    get_polynomial_decay_schedule_with_warmup,
)
from transformers.trainer_pt_utils import get_tpu_sampler
from transformers.training_args import ParallelMode
from transformers.utils import is_torch_tpu_available


if is_fairscale_available():
				from fairscale.optim import OSS


__snake_case								=  logging.get_logger(__name__)

__snake_case								=  {
    """linear""": get_linear_schedule_with_warmup,
    """cosine""": get_cosine_schedule_with_warmup,
    """cosine_w_restarts""": get_cosine_with_hard_restarts_schedule_with_warmup,
    """polynomial""": get_polynomial_decay_schedule_with_warmup,
    """constant""": get_constant_schedule,
    """constant_w_warmup""": get_constant_schedule_with_warmup,
}




class 						lowercase__  (						_UpperCAmelCase  ):
	def __init__(			self							:   str       ,   UpperCAmelCase_							:   int=None       ,   UpperCAmelCase_							:   List[str]=None       ,   *UpperCAmelCase_							:   Union[str, Any]       ,   **UpperCAmelCase_							:   Optional[Any]							):
						super().__init__(*UpperCAmelCase_       ,   **UpperCAmelCase_							)

						if config is None:
											assert isinstance(self.model       ,   UpperCAmelCase_							), (
											    "If no `config` is passed the model to be trained has to be of type `PreTrainedModel`, but is"
											    F' {self.model.__class__}'
											)
											SCREAMING_SNAKE_CASE__													= self.model.config
						else:
											SCREAMING_SNAKE_CASE__													= config

						SCREAMING_SNAKE_CASE__													= data_args
						SCREAMING_SNAKE_CASE__													= self.config.tgt_vocab_size if isinstance(self.config       ,   UpperCAmelCase_							) else self.config.vocab_size

						if self.args.label_smoothing != 0 or (self.data_args is not None and self.data_args.ignore_pad_token_for_loss):
											assert self.config.pad_token_id is not None, (
											    "Make sure that `config.pad_token_id` is correcly defined when ignoring `pad_token` for loss"
											    " calculation or doing label smoothing."
											)

						if self.config.pad_token_id is None and self.config.eos_token_id is not None:
											logger.warning(
											    F'The `config.pad_token_id` is `None`. Using `config.eos_token_id` = {self.config.eos_token_id} for'
											    ' padding..'							)

						if self.args.label_smoothing == 0:
											SCREAMING_SNAKE_CASE__													= torch.nn.CrossEntropyLoss(ignore_index=self.config.pad_token_id							)
						else:
											# dynamically import label_smoothed_nll_loss
											from utils import label_smoothed_nll_loss

											SCREAMING_SNAKE_CASE__													= label_smoothed_nll_loss
	def      A_				(			self							:   Tuple       ,   UpperCAmelCase_							:   int							):
						if self.optimizer is None:
											SCREAMING_SNAKE_CASE__													= ['bias', 'LayerNorm.weight']
											SCREAMING_SNAKE_CASE__													= [
											    {
											        'params': [p for n, p in self.model.named_parameters() if not any(nd in n for nd in no_decay							)],
											        'weight_decay': self.args.weight_decay,
											    },
											    {
											        'params': [p for n, p in self.model.named_parameters() if any(nd in n for nd in no_decay							)],
											        'weight_decay': 0.0,
											    },
											]
											SCREAMING_SNAKE_CASE__													= Adafactor if self.args.adafactor else AdamW
											if self.args.adafactor:
																SCREAMING_SNAKE_CASE__													= Adafactor
																SCREAMING_SNAKE_CASE__													= {'scale_parameter': False, 'relative_step': False}
											else:
																SCREAMING_SNAKE_CASE__													= AdamW
																SCREAMING_SNAKE_CASE__													= {
																    'betas': (self.args.adam_betaa, self.args.adam_betaa),
																    'eps': self.args.adam_epsilon,
																}
											SCREAMING_SNAKE_CASE__													= self.args.learning_rate
											if self.sharded_ddp:
																SCREAMING_SNAKE_CASE__													= OSS(
																    params=UpperCAmelCase_       ,   optim=UpperCAmelCase_       ,   **UpperCAmelCase_       ,   )
											else:
																SCREAMING_SNAKE_CASE__													= optimizer_cls(UpperCAmelCase_       ,   **UpperCAmelCase_							)

						if self.lr_scheduler is None:
											SCREAMING_SNAKE_CASE__													= self._get_lr_scheduler(UpperCAmelCase_							)
						else:  # ignoring --lr_scheduler
											logger.warning('scheduler is passed to `Seq2SeqTrainer`, `--lr_scheduler` arg is ignored.'							)
	def      A_				(			self							:   str       ,   UpperCAmelCase_							:   List[Any]							):
						SCREAMING_SNAKE_CASE__													= arg_to_scheduler[self.args.lr_scheduler]
						if self.args.lr_scheduler == "constant":
											SCREAMING_SNAKE_CASE__													= schedule_func(self.optimizer							)
						elif self.args.lr_scheduler == "constant_w_warmup":
											SCREAMING_SNAKE_CASE__													= schedule_func(self.optimizer       ,   num_warmup_steps=self.args.warmup_steps							)
						else:
											SCREAMING_SNAKE_CASE__													= schedule_func(
											    self.optimizer       ,   num_warmup_steps=self.args.warmup_steps       ,   num_training_steps=UpperCAmelCase_							)
						return scheduler
	def      A_				(			self							:   List[str]							):
						if isinstance(self.train_dataset       ,   torch.utils.data.IterableDataset							):
											return None
						elif is_torch_tpu_available():
											return get_tpu_sampler(self.train_dataset							)
						else:
											if self.args.sortish_sampler:
																self.train_dataset.make_sortish_sampler(
																    self.args.per_device_train_batch_size       ,   distributed=(self.args.parallel_mode == ParallelMode.DISTRIBUTED)       ,   )

											return (
											    RandomSampler(self.train_dataset							)
											    if self.args.local_rank == -1
											    else DistributedSampler(self.train_dataset							)
											)
	def      A_				(			self							:   Any       ,   UpperCAmelCase_							:   Tuple       ,   UpperCAmelCase_							:   Optional[Any]       ,   UpperCAmelCase_							:   Dict							):
						if self.args.label_smoothing == 0:
											if self.data_args is not None and self.data_args.ignore_pad_token_for_loss:
																# force training to ignore pad token
																SCREAMING_SNAKE_CASE__													= model(**UpperCAmelCase_       ,   use_cache=UpperCAmelCase_							)[0]
																SCREAMING_SNAKE_CASE__													= self.loss_fn(logits.view(-1       ,   logits.shape[-1]							)       ,   labels.view(-1							)							)
											else:
																# compute usual loss via models
																SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= model(**UpperCAmelCase_       ,   labels=UpperCAmelCase_       ,   use_cache=UpperCAmelCase_							)[:2]
						else:
											# compute label smoothed loss
											SCREAMING_SNAKE_CASE__													= model(**UpperCAmelCase_       ,   use_cache=UpperCAmelCase_							)[0]
											SCREAMING_SNAKE_CASE__													= torch.nn.functional.log_softmax(UpperCAmelCase_       ,   dim=-1							)
											SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= self.loss_fn(UpperCAmelCase_       ,   UpperCAmelCase_       ,   self.args.label_smoothing       ,   ignore_index=self.config.pad_token_id							)
						return loss, logits
	def      A_				(			self							:   Union[str, Any]       ,   UpperCAmelCase_							:   str       ,   UpperCAmelCase_							:   List[Any]							):
						SCREAMING_SNAKE_CASE__													= inputs.pop('labels'							)
						SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= self._compute_loss(UpperCAmelCase_       ,   UpperCAmelCase_       ,   UpperCAmelCase_							)
						return loss
	def      A_				(			self							:   List[str]       ,   UpperCAmelCase_							:   nn.Module       ,   UpperCAmelCase_							:   Dict[str, Union[torch.Tensor, Any]]       ,   UpperCAmelCase_							:   bool       ,   UpperCAmelCase_							:   Optional[List[str]] = None       ,   ):
						SCREAMING_SNAKE_CASE__													= self._prepare_inputs(UpperCAmelCase_							)

						SCREAMING_SNAKE_CASE__													= {
						    'max_length': self.data_args.val_max_target_length
						    if self.data_args is not None
						    else self.config.max_length,
						    'num_beams': self.data_args.eval_beams if self.data_args is not None else self.config.num_beams,
						}

						if self.args.predict_with_generate and not self.args.prediction_loss_only:
											SCREAMING_SNAKE_CASE__													= self.model.generate(
											    inputs['input_ids']       ,   attention_mask=inputs['attention_mask']       ,   **UpperCAmelCase_       ,   )
											# in case the batch is shorter than max length, the output should be padded
											if generated_tokens.shape[-1] < gen_kwargs["max_length"]:
																SCREAMING_SNAKE_CASE__													= self._pad_tensors_to_max_len(UpperCAmelCase_       ,   gen_kwargs['max_length']							)

						SCREAMING_SNAKE_CASE__													= inputs.pop('labels'							)
						with torch.no_grad():
											# compute loss on predict data
											SCREAMING_SNAKE_CASE__						, SCREAMING_SNAKE_CASE__													= self._compute_loss(UpperCAmelCase_       ,   UpperCAmelCase_       ,   UpperCAmelCase_							)

						SCREAMING_SNAKE_CASE__													= loss.mean().detach()
						if self.args.prediction_loss_only:
											return (loss, None, None)

						SCREAMING_SNAKE_CASE__													= generated_tokens if self.args.predict_with_generate else logits

						if labels.shape[-1] < gen_kwargs["max_length"]:
											SCREAMING_SNAKE_CASE__													= self._pad_tensors_to_max_len(UpperCAmelCase_       ,   gen_kwargs['max_length']							)

						return (loss, logits, labels)





	def      A_				(			self							:   Union[str, Any]       ,   UpperCAmelCase_							:   Tuple       ,   UpperCAmelCase_							:   List[str]							):
						# If PAD token is not defined at least EOS token has to be defined
						SCREAMING_SNAKE_CASE__													= self.config.pad_token_id if self.config.pad_token_id is not None else self.config.eos_token_id

						if pad_token_id is None:
											raise ValueError(
											    'Make sure that either `config.pad_token_id` or `config.eos_token_id` is defined if tensor has to be'
											    F' padded to `max_length`={max_length}'							)

						SCREAMING_SNAKE_CASE__													= pad_token_id * torch.ones(
						    (tensor.shape[0], max_length)       ,   dtype=tensor.dtype       ,   device=tensor.device							)
						SCREAMING_SNAKE_CASE__													= tensor
						return padded_tensor





