



def       A_   (						__a  :					list		):

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

def       A_   (						__a  :					list		):

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


if __name__ == "__main__":
	import doctest

	doctest.testmod()






import math
from decimal import Decimal, getcontext

def       A_   (						__a  :					int = 4_000_000		):

				"""simple docstring"""

				try:
								a__					   =		int(__a		)
				except (TypeError, ValueError):
								raise TypeError("""Parameter n must be int or castable to int."""		)
				if n <= 0:
								raise ValueError("""Parameter n must be greater than or equal to one."""		)
				a__					   =		100
				a__					   =		(Decimal(5		) ** Decimal(0.5		) + 1) / Decimal(2		)

				a__					   =		(math.floor(math.log(n * (phi + 2)			,		__a		) - 1		) // 3) * 3 + 2
				a__					   =		Decimal(round(phi ** Decimal(index + 1		)		)		) / (phi + 2)
				a__					   =		num // 2
				return int(__a		)


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

