code: 
  import math
  from itertools import permutations
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						int     )       ->	bool:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	if 1 < number < 4:
  		# 2 and 3 are primes
  		return True
  	elif number < 2 or number % 2 == 0 or number % 3 == 0:
  		# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
  		return False
  
  	# All primes number are in format of 6k +/- 1
  	for i in range(5   ,     int(math.sqrt(_snake_case     ) + 1     )   ,     6     ):
  		if number % i == 0 or number % (i + 2) == 0:
  			return False
  	return True
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						int   ,     _snake_case   :						list     )       ->	bool:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__     ,					__magic_name__	:	int			     =  0, len(_snake_case     ) - 1
  	while left <= right:
  		__magic_name__	:	Union[str, Any]			     =  (left + right) // 2
  		if prime_list[middle] == target:
  			return True
  		elif prime_list[middle] < target:
  			__magic_name__	:	Dict			     =  middle + 1
  		else:
  			__magic_name__	:	Optional[Any]			     =  middle - 1
  
  	return False
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      )       ->	Dict:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	Dict			     =  [n for n in range(1001   ,     10000   ,     2     ) if is_prime(_snake_case     )]
  	__magic_name__	:	int			     =  []
  
  	for number in prime_list:
  		__magic_name__	:	str			     =  []
  
  		for prime_member in permutations(list(str(_snake_case     )     )     ):
  			__magic_name__	:	str			     =  int("".join(_snake_case     )     )
  
  			if prime % 2 == 0:
  				continue
  
  			if search(_snake_case   ,     _snake_case     ):
  				tmp_numbers.append(_snake_case     )
  
  		tmp_numbers.sort()
  		if len(_snake_case     ) >= 3:
  			candidates.append(_snake_case     )
  
  	__magic_name__	:	Optional[int]			     =  []
  	for candidate in candidates:
  		__magic_name__	:	List[str]			     =  len(_snake_case     )
  		__magic_name__	:	List[str]			     =  False
  
  		for i in range(_snake_case     ):
  			for j in range(i + 1   ,     _snake_case     ):
  				for k in range(j + 1   ,     _snake_case     ):
  					if (
  					    abs(candidate[i] - candidate[j]     )
  					    == abs(candidate[j] - candidate[k]     )
  					    and len({candidate[i], candidate[j], candidate[k]}     ) == 3
  					):
  						passed.append(
  						    sorted([candidate[i], candidate[j], candidate[k]]     )     )
  						__magic_name__	:	Optional[Any]			     =  True
  
  					if found:
  						break
  				if found:
  					break
  			if found:
  				break
  
  	__magic_name__	:	Tuple			     =  set()
  	for seq in passed:
  		answer.add("".join([str(_snake_case     ) for i in seq]     )     )
  
  	return max(int(_snake_case     ) for x in answer     )
  
  
  if __name__ == "__main__":
  				print(solution())
  
style_context: 
  import argparse
  import gc
  import os
  import threading
  
  import evaluate
  import psutil
  import torch
  from datasets import load_dataset
  from torch.distributed.fsdp.fully_sharded_data_parallel import FullOptimStateDictConfig, FullStateDictConfig
  from torch.utils.data import DataLoader
  from transformers import (
      AutoModelForSequenceClassification,
      AutoTokenizer,
      get_linear_schedule_with_warmup,
      set_seed,
  )
  
  from accelerate import Accelerator, DistributedType, FullyShardedDataParallelPlugin
  
  
  ########################################################################
  # This is a fully working simple example to use Accelerate
  #
  # This example trains a Bert base model on GLUE MRPC
  # in any of the following settings (with the same script):
  #   - single CPU or single GPU
  #   - multi GPUS (using PyTorch distributed mode)
  #   - (multi) TPUs
  #   - fp16 (mixed-precision) or fp32 (normal precision)
  #   - FSDP
  #
  # This example also demonstrates the checkpointing and sharding capabilities
  #
  # To run it in each of these various modes, follow the instructions
  # in the readme for examples:
  # https://github.com/huggingface/accelerate/tree/main/examples
  #
  ########################################################################
  
  
  snake_case				:	Any							   =						16
  snake_case				:	Union[str, Any]							   =						32
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						int     )       ->	Any:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	return int(x / 2**20     )
  
  
  
  
  
  class    _snake_case      :
  
  
  
  
  
  
  
  			def __enter__(       self	):
  				gc.collect()
  				torch.cuda.empty_cache()
  				torch.cuda.reset_max_memory_allocated()  # reset the peak gauge to zero
  				__magic_name__	:	List[Any]			     =  torch.cuda.memory_allocated()
  				__magic_name__	:	Dict			     =  psutil.Process()
  
  				__magic_name__	:	List[str]			     =  self.cpu_mem_used()
  				__magic_name__	:	Optional[int]			     =  True
  				__magic_name__	:	int			     =  threading.Thread(target=self.peak_monitor_func	)
  				__magic_name__	:	str			     =  True
  				peak_monitor_thread.start()
  				return self
  
  
  
  
  
  
  
  			def     SCREAMING_SNAKE_CASE					(       self	):
  				return self.process.memory_info().rss
  
  
  
  
  
  
  
  			def     SCREAMING_SNAKE_CASE					(       self	):
  				__magic_name__	:	Dict			     =  -1
  
  				while True:
  					__magic_name__	:	Optional[Any]			     =  max(self.cpu_mem_used()		,     self.cpu_peak	)
  
  					# can't sleep or will not catch the peak right (this comment is here on purpose)
  					# time.sleep(0.001) # 1msec
  
  					if not self.peak_monitoring:
  						break
  
  
  
  
  
  
  
  			def __exit__(       self		,     *_a	):
  				__magic_name__	:	List[Any]			     =  False
  
  				gc.collect()
  				torch.cuda.empty_cache()
  				__magic_name__	:	Union[str, Any]			     =  torch.cuda.memory_allocated()
  				__magic_name__	:	Any			     =  torch.cuda.max_memory_allocated()
  				__magic_name__	:	List[Any]			     =  bamb(self.end - self.begin	)
  				__magic_name__	:	List[str]			     =  bamb(self.peak - self.begin	)
  
  				__magic_name__	:	List[str]			     =  self.cpu_mem_used()
  				__magic_name__	:	int			     =  bamb(self.cpu_end - self.cpu_begin	)
  				__magic_name__	:	List[Any]			     =  bamb(self.cpu_peak - self.cpu_begin	)
  				# print(f"delta used/peak {self.used:4d}/{self.peaked:4d}")
  
  
  # For testing only
  if os.environ.get("TESTING_MOCKED_DATALOADERS", None) == "1":
  				from accelerate.test_utils.training import mocked_dataloaders
  
  				snake_case				:	Tuple							   =						mocked_dataloaders  # noqa: F811
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      _snake_case   :						str   ,     _snake_case   :						Dict     )       ->	Dict:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	if os.environ.get("TESTING_MOCKED_DATALOADERS"   ,     _snake_case     ) == "1":
  		__magic_name__	:	Optional[Any]			     =  2
  
  	# New Code #
  	# Pass the advanced FSDP settings not part of the accelerate config by creating fsdp_plugin
  	__magic_name__	:	Dict			     =  FullyShardedDataParallelPlugin(
  	    state_dict_config=FullStateDictConfig(offload_to_cpu=_snake_case   ,     ranka_only=_snake_case     )   ,     optim_state_dict_config=FullOptimStateDictConfig(offload_to_cpu=_snake_case   ,     ranka_only=_snake_case     )   ,     )
  
  	# Initialize accelerator
  	if args.with_tracking:
  		__magic_name__	:	Union[str, Any]			     =  Accelerator(
  		    cpu=args.cpu   ,     mixed_precision=args.mixed_precision   ,     log_with="wandb"   ,     project_dir=args.logging_dir   ,     fsdp_plugin=_snake_case   ,     )
  	else:
  		__magic_name__	:	List[str]			     =  Accelerator(fsdp_plugin=_snake_case     )
  	accelerator.print(accelerator.distributed_type     )
  
  	if hasattr(args.checkpointing_steps   ,     "isdigit"     ):
  		if args.checkpointing_steps == "epoch":
  			__magic_name__	:	List[Any]			     =  args.checkpointing_steps
  		elif args.checkpointing_steps.isdigit():
  			__magic_name__	:	List[Any]			     =  int(args.checkpointing_steps     )
  		else:
  			raise ValueError(
  			    F'''Argument `checkpointing_steps` must be either a number or `epoch`. `{args.checkpointing_steps}` passed.'''     )
  	else:
  		__magic_name__	:	List[str]			     =  None
  	# Sample hyper-parameters for learning rate, batch size, seed and a few other HPs
  	__magic_name__	:	List[Any]			     =  config["lr"]
  	__magic_name__	:	Optional[Any]			     =  int(config["num_epochs"]     )
  	__magic_name__	:	Optional[int]			     =  int(config["seed"]     )
  	__magic_name__	:	int			     =  int(config["batch_size"]     )
  
  	# We need to initialize the trackers we use, and also store our configuration
  	if args.with_tracking:
  		__magic_name__	:	List[Any]			     =  vars(_snake_case     )
  		accelerator.init_trackers("fsdp_glue_no_trainer"   ,     _snake_case     )
  
  	__magic_name__	:	Dict			     =  AutoTokenizer.from_pretrained(args.model_name_or_path     )
  	__magic_name__	:	List[Any]			     =  load_dataset("glue"   ,     "mrpc"     )
  	__magic_name__	:	Union[str, Any]			     =  evaluate.load("glue"   ,     "mrpc"     )
  
  	def tokenize_function(_snake_case   :						List[Any]     ):
  		# max_length=None => use the model max length (it's actually the default)
  		__magic_name__	:	Union[str, Any]			     =  tokenizer(examples["sentence1"]   ,     examples["sentence2"]   ,     truncation=_snake_case   ,     max_length=_snake_case     )
  		return outputs
  
  	# Apply the method we just defined to all the examples in all the splits of the dataset
  	# starting with the main process first:
  	with accelerator.main_process_first():
  		__magic_name__	:	int			     =  datasets.map(
  		    _snake_case   ,     batched=_snake_case   ,     remove_columns=["idx", "sentence1", "sentence2"]   ,     )
  
  	# We also rename the 'label' column to 'labels' which is the expected name for labels by the models of the
  	# transformers library
  	__magic_name__	:	Optional[Any]			     =  tokenized_datasets.rename_column("label"   ,     "labels"     )
  
  	# If the batch size is too big we use gradient accumulation
  	__magic_name__	:	Dict			     =  1
  	if batch_size > MAX_GPU_BATCH_SIZE and accelerator.distributed_type != DistributedType.TPU:
  		__magic_name__	:	List[Any]			     =  batch_size // MAX_GPU_BATCH_SIZE
  		__magic_name__	:	Dict			     =  MAX_GPU_BATCH_SIZE
  
  	def collate_fn(_snake_case   :						Union[str, Any]     ):
  		# On TPU it's best to pad everything to the same length or training will be very slow.
  		__magic_name__	:	Tuple			     =  128 if accelerator.distributed_type == DistributedType.TPU else None
  		# When using mixed precision we want round multiples of 8/16
  		if accelerator.mixed_precision == "fp8":
  			__magic_name__	:	List[str]			     =  16
  		elif accelerator.mixed_precision != "no":
  			__magic_name__	:	Any			     =  8
  		else:
  			__magic_name__	:	Tuple			     =  None
  
  		return tokenizer.pad(
  		    _snake_case   ,     padding="longest"   ,     max_length=_snake_case   ,     pad_to_multiple_of=_snake_case   ,     return_tensors="pt"   ,     )
  
  	# Instantiate dataloaders.
  	__magic_name__	:	Optional[int]			     =  DataLoader(
  	    tokenized_datasets["train"]   ,     shuffle=_snake_case   ,     collate_fn=_snake_case   ,     batch_size=_snake_case     )
  	__magic_name__	:	Dict			     =  DataLoader(
  	    tokenized_datasets["validation"]   ,     shuffle=_snake_case   ,     collate_fn=_snake_case   ,     batch_size=_snake_case     )
  
  	set_seed(_snake_case     )
  
  	# Instantiate the model (we build the model here so that the seed also control new weights initialization)
  	__magic_name__	:	Optional[Any]			     =  AutoModelForSequenceClassification.from_pretrained(
  	    args.model_name_or_path   ,     return_dict=_snake_case   ,     low_cpu_mem_usage=_snake_case     )
  
  	# New Code #
  	# For FSDP feature, it is highly recommended and efficient to prepare the model before creating optimizer
  	__magic_name__	:	List[str]			     =  accelerator.prepare(_snake_case     )
  	accelerator.print(_snake_case     )
  
  	# Instantiate optimizer
  	# New Code #
  	# For FSDP feature, at present it doesn't support multiple parameter groups,
  	# so we need to create a single parameter group for the whole model
  	__magic_name__	:	Optional[Any]			     =  torch.optim.AdamW(params=model.parameters()   ,     lr=_snake_case   ,     weight_decay=2E-4     )
  
  	# Instantiate scheduler
  	__magic_name__	:	Any			     =  get_linear_schedule_with_warmup(
  	    optimizer=_snake_case   ,     num_warmup_steps=10   ,     num_training_steps=(len(_snake_case     ) * num_epochs) // gradient_accumulation_steps   ,     )
  
  	# New Code #
  	# For FSDP feature, prepare everything except the model as we have already prepared the model
  	# before creating the optimizer
  	# There is no specific order to remember, we just need to unpack the objects in the same order we gave them to the
  	# prepare method.
  	__magic_name__     ,					__magic_name__     ,					__magic_name__     ,					__magic_name__	:	Dict			     =  accelerator.prepare(
  	    _snake_case   ,     _snake_case   ,     _snake_case   ,     _snake_case     )
  
  	__magic_name__	:	Union[str, Any]			     =  0
  
  	# Potentially load in the weights and states from a previous save
  	if args.resume_from_checkpoint:
  		if args.resume_from_checkpoint is not None or args.resume_from_checkpoint != "":
  			accelerator.print(F'''Resumed from checkpoint: {args.resume_from_checkpoint}'''     )
  			accelerator.load_state(args.resume_from_checkpoint     )
  			__magic_name__	:	Optional[Any]			     =  os.path.basename(args.resume_from_checkpoint     )
  		else:
  			# Get the most recent checkpoint
  			__magic_name__	:	Tuple			     =  [f.name for f in os.scandir(os.getcwd()     ) if f.is_dir()]
  			dirs.sort(key=os.path.getctime     )
  			__magic_name__	:	Tuple			     =  dirs[-1]  # Sorts folders by date modified, most recent checkpoint is the last
  		# Extract `epoch_{i}` or `step_{i}`
  		__magic_name__	:	Dict			     =  os.path.splitext(_snake_case     )[0]
  
  		if "epoch" in training_difference:
  			num_epochs -= int(training_difference.replace("epoch_"   ,     ""     )     )
  			__magic_name__	:	List[Any]			     =  None
  		else:
  			__magic_name__	:	List[str]			     =  int(training_difference.replace("step_"   ,     ""     )     )
  			num_epochs -= resume_step // len(_snake_case     )
  			# If resuming by step, we also need to know exactly how far into the DataLoader we went
  			__magic_name__	:	Tuple			     =  (num_epochs * len(_snake_case     )) - resume_step
  
      # Now we train the model
  	for epoch in range(_snake_case     ):
  		# New Code #
  		# context manager to track the peak memory usage during the training epoch
  		with TorchTracemalloc() as tracemalloc:
  			model.train()
  			if args.with_tracking:
  				__magic_name__	:	List[Any]			     =  0
  			for step, batch in enumerate(_snake_case     ):
  				# We need to skip steps until we reach the resumed step
  				if args.resume_from_checkpoint and epoch == 0:
  					if resume_step is not None and step < resume_step:
  						pass
                  # We could avoid this line since we set the accelerator with `device_placement=True`.
  				batch.to(accelerator.device     )
  				__magic_name__	:	List[Any]			     =  model(**_snake_case     )
  				__magic_name__	:	Optional[int]			     =  outputs.loss
  				__magic_name__	:	Tuple			     =  loss / gradient_accumulation_steps
  				# We keep track of the loss at each epoch
  				if args.with_tracking:
  					total_loss += loss.detach().float()
  				accelerator.backward(_snake_case     )
  				if step % gradient_accumulation_steps == 0:
  					optimizer.step()
  					lr_scheduler.step()
  					optimizer.zero_grad()
  					# accelerator.print(lr_scheduler.get_lr())
  
  				overall_step += 1
  
  				if isinstance(_snake_case   ,     _snake_case     ):
  					__magic_name__	:	List[Any]			     =  F'''step_{overall_step}'''
  					if overall_step % checkpointing_steps == 0:
  						if args.output_dir is not None:
  							__magic_name__	:	List[str]			     =  os.path.join(args.output_dir   ,     _snake_case     )
  						accelerator.save_state(_snake_case     )
          # New Code #
          # Printing the GPU memory usage details such as allocated memory, peak memory, and total memory usage
  		accelerator.print("Memory before entering the train : {}".format(bamb(tracemalloc.begin     )     )     )
  		accelerator.print("Memory consumed at the end of the train (end-begin): {}".format(tracemalloc.used     )     )
  		accelerator.print("Peak Memory consumed during the train (max-begin): {}".format(tracemalloc.peaked     )     )
  		accelerator.print(
  		    "Total Peak Memory consumed during the train (max): {}".format(
  		        tracemalloc.peaked + bamb(tracemalloc.begin     )     )     )
  		# Logging the peak memory usage of the GPU to the tracker
  		if args.with_tracking:
  			accelerator.log(
  			    {
  			        "train_total_peak_memory": tracemalloc.peaked + bamb(tracemalloc.begin     ),
  			    }   ,     step=_snake_case   ,     )
  
  		# New Code #
  		# context manager to track the peak memory usage during the evaluation
  		with TorchTracemalloc() as tracemalloc:
  			model.eval()
  			for step, batch in enumerate(_snake_case     ):
  				# We could avoid this line since we set the accelerator with `device_placement=True`.
  				batch.to(accelerator.device     )
  				with torch.no_grad():
  					__magic_name__	:	Tuple			     =  model(**_snake_case     )
  				__magic_name__	:	Any			     =  outputs.logits.argmax(dim=-1     )
  				__magic_name__     ,					__magic_name__	:	Any			     =  accelerator.gather_for_metrics((predictions, batch["labels"])     )
  				metric.add_batch(
  				    predictions=_snake_case   ,     references=_snake_case   ,     )
  
  			__magic_name__	:	Tuple			     =  metric.compute()
  			# Use accelerator.print to print only on the main process.
  			accelerator.print(F'''epoch {epoch}:'''   ,     _snake_case     )
  			if args.with_tracking:
  				accelerator.log(
  				    {
  				        "accuracy": eval_metric["accuracy"],
  				        "f1": eval_metric["f1"],
  				        "train_loss": total_loss.item() / len(_snake_case     ),
  				    }   ,     step=_snake_case   ,     )
  
  			if checkpointing_steps == "epoch":
  				__magic_name__	:	Dict			     =  F'''epoch_{epoch}'''
  				if args.output_dir is not None:
  					__magic_name__	:	Dict			     =  os.path.join(args.output_dir   ,     _snake_case     )
  				accelerator.save_state(_snake_case     )
          # New Code #
          # Printing the GPU memory usage details such as allocated memory, peak memory, and total memory usage
  		accelerator.print("Memory before entering the eval : {}".format(bamb(tracemalloc.begin     )     )     )
  		accelerator.print("Memory consumed at the end of the eval (end-begin): {}".format(tracemalloc.used     )     )
  		accelerator.print("Peak Memory consumed during the eval (max-begin): {}".format(tracemalloc.peaked     )     )
  		accelerator.print(
  		    "Total Peak Memory consumed during the eval (max): {}".format(tracemalloc.peaked + bamb(tracemalloc.begin     )     )     )
  		# Logging the peak memory usage of the GPU to the tracker
  		if args.with_tracking:
  			accelerator.log(
  			    {
  			        "eval_total_peak_memory": tracemalloc.peaked + bamb(tracemalloc.begin     ),
  			    }   ,     step=_snake_case   ,     )
  
  	if args.with_tracking:
  		accelerator.end_training()
  
  
  
  
  
  
  def 						lowerCAmelCase_		(      )       ->	Optional[int]:
  
  
  
  
  
  
  
  	'''simple docstring'''
  
  
  	__magic_name__	:	List[str]			     =  argparse.ArgumentParser(description="Simple example of training script."     )
  	parser.add_argument(
  	    "--mixed_precision"   ,     type=_snake_case   ,     default=_snake_case   ,     choices=["no", "fp16", "bf16", "fp8"]   ,     help="Whether to use mixed precision. Choose"
  	    "between fp16 and bf16 (bfloat16). Bf16 requires PyTorch >= 1.10."
  	    "and an Nvidia Ampere GPU."   ,     )
  	parser.add_argument("--cpu"   ,     action="store_true"   ,     help="If passed, will train on the CPU."     )
  	parser.add_argument(
  	    "--checkpointing_steps"   ,     type=_snake_case   ,     default=_snake_case   ,     help="Whether the various states should be saved at the end of every n steps, or 'epoch' for each epoch."   ,     )
  	parser.add_argument(
  	    "--resume_from_checkpoint"   ,     type=_snake_case   ,     default=_snake_case   ,     help="If the training should continue from a checkpoint folder."   ,     )
  	parser.add_argument(
  	    "--with_tracking"   ,     action="store_true"   ,     help="Whether to load in all available experiment trackers from the environment and use them for logging."   ,     )
  	parser.add_argument(
  	    "--output_dir"   ,     type=_snake_case   ,     default="."   ,     help="Optional save directory where all checkpoint folders will be stored. Default is the current working directory."   ,     )
  	parser.add_argument(
  	    "--logging_dir"   ,     type=_snake_case   ,     default="logs"   ,     help="Location on where to store experiment tracking logs`"   ,     )
  	parser.add_argument(
  	    "--model_name_or_path"   ,     type=_snake_case   ,     help="Path to pretrained model or model identifier from huggingface.co/models."   ,     required=_snake_case   ,     )
  	__magic_name__	:	str			     =  parser.parse_args()
  	__magic_name__	:	Dict			     =  {"lr": 2E-5, "num_epochs": 3, "seed": 42, "batch_size": 16}
  	training_function(_snake_case   ,     _snake_case     )
  
  
  if __name__ == "__main__":
  				main()
  