def     A__	(       __A						):







    '''simple docstring'''
    return n * (2 * n - 1)







def     A__	(       __A						):







    '''simple docstring'''
    _lowerCamelCase  :		Union[str, Any]			    =    (1 + 24 * n) ** 0.5
    return ((1 + root) / 6) % 1 == 0







def     A__	(       __A = 144						):







    '''simple docstring'''
    _lowerCamelCase  :		int			    =    start
    _lowerCamelCase  :		str			    =    hexagonal_num(__A						)
    while not is_pentagonal(__A						):
        n += 1
        _lowerCamelCase  :		List[Any]			    =    hexagonal_num(__A						)
    return num


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





import torch

from accelerate import Accelerator, DistributedType




class 						__snake_case						:





      '''simple docstring'''






      def __enter__(       self  :					Union[str, Any])					->Dict:


          """simple docstring"""


          if self.enabled:
              _lowerCamelCase  :		str			    =    self.model.no_sync()
              self.model_sync_obj.__enter__()

          return self



      def __exit__(       self  :					List[str]     ,	_UpperCamelCase  :					Any     ,	_UpperCamelCase  :					Union[str, Any]     ,	_UpperCamelCase  :					int)					->Optional[Any]:


          """simple docstring"""


          if self.enabled:
              # Average all models on exit
              self._sync_and_avg_model_params()
              self.model_sync_obj.__exit__(_UpperCamelCase     ,	_UpperCamelCase     ,	_UpperCamelCase)



      def __init__(       self  :					Union[str, Any]     ,	_UpperCamelCase  :					Accelerator     ,	_UpperCamelCase  :					torch.nn.Module     ,	_UpperCamelCase  :					int     ,	_UpperCamelCase  :					bool = True)					->int:


          """simple docstring"""


          if accelerator.distributed_type not in [
              DistributedType.NO,
              DistributedType.MULTI_CPU,
              DistributedType.MULTI_GPU,
          ]:
              raise NotImplementedError("""LocalSGD is supported only for CPUs and GPUs (no DeepSpeed or MegatronLM)""")
          _lowerCamelCase  :		str			    =    enabled and accelerator.distributed_type != DistributedType.NO
          _lowerCamelCase  :		Optional[int]			    =    0
          if self.enabled:
              _lowerCamelCase  :		Union[str, Any]			    =    accelerator
              _lowerCamelCase  :		int			    =    model
              _lowerCamelCase  :		int			    =    local_sgd_steps



      def 						_SCREAMING_SNAKE_CASE     (       self  :					Union[str, Any])					->Union[str, Any]:


          """simple docstring"""


          self.num_steps += 1
          if not self.enabled:
              return

          if self.num_steps % self.local_sgd_steps == 0:
              self._sync_and_avg_model_params()




      def 						_SCREAMING_SNAKE_CASE     (       self  :					Optional[Any])					->List[str]:


          """simple docstring"""


          self.accelerator.wait_for_everyone()
          with self.accelerator.autocast():
              for param in self.model.parameters():
                  _lowerCamelCase  :		Union[str, Any]			    =    self.accelerator.reduce(param.data     ,	reduction="""mean""")




