




"""simple docstring"""







def 							lowercase   (snake_case__				:	list[int | list] ,    snake_case__				:	int	)       ->					int:


				'''simple docstring'''






				lowerCAmelCase		=			0
				for ele in arr:
								total_sum += product_sum(snake_case__ ,    depth + 1	) if isinstance(snake_case__ ,    snake_case__	) else ele
				return total_sum * depth





def 							lowercase   (snake_case__				:	list[int | list]	)       ->					int:


				'''simple docstring'''






				return product_sum(snake_case__ ,    1	)


if __name__ == "__main__":
		import doctest

		doctest.testmod()











"""simple docstring"""







def 							lowercase   (snake_case__				:	list	)       ->					bool:


				'''simple docstring'''






				if not isinstance(snake_case__ ,    snake_case__	):
								raise ValueError("""Input series is not valid, valid series - [2, 4, 8]"""	)
				if len(snake_case__	) == 0:
								raise ValueError("""Input list must be a non empty list"""	)
				if len(snake_case__	) == 1:
								return True
				try:
								lowerCAmelCase		=			series[1] / series[0]
								for index in range(len(snake_case__	) - 1	):
												if series[index + 1] / series[index] != common_ratio:
																return False
				except ZeroDivisionError:
								return False
				return True





def 							lowercase   (snake_case__				:	list	)       ->					float:


				'''simple docstring'''






				if not isinstance(snake_case__ ,    snake_case__	):
								raise ValueError("""Input series is not valid, valid series - [2, 4, 8]"""	)
				if len(snake_case__	) == 0:
								raise ValueError("""Input list must be a non empty list"""	)
				lowerCAmelCase		=			1
				for value in series:
								answer *= value
				return pow(snake_case__ ,    1 / len(snake_case__	)	)


if __name__ == "__main__":
		import doctest

		doctest.testmod()





