






def __A					(   __lowerCAmelCase			)->						int:





						"""simple docstring"""



						if not isinstance(__lowerCAmelCase				,	__lowerCAmelCase			) or not all(
						    isinstance(__lowerCAmelCase				,	__lowerCAmelCase			) and item >= 0 for item in citations			):
												raise ValueError('The citations should be a list of non negative integers.'			)

						citations.sort()
						_UpperCAmelCase             =    len(__lowerCAmelCase			)

						for i in range(__lowerCAmelCase			):
												if citations[len_citations - 1 - i] <= i:
																		return i

						return len_citations


if __name__ == "__main__":
		import doctest

		doctest.testmod()








import math
from timeit import timeit


def __A					(   __lowerCAmelCase			)->						int:





						"""simple docstring"""



						_UpperCAmelCase             =    0
						_UpperCAmelCase             =    abs(__lowerCAmelCase			)
						while True:
												_UpperCAmelCase             =    n // 10
												digits += 1
												if n == 0:
																		break
						return digits


def __A					(   __lowerCAmelCase			)->						int:





						"""simple docstring"""



						return 1 if n == 0 else math.floor(math.log(abs(__lowerCAmelCase			)				,	10			) + 1			)


def __A					(   __lowerCAmelCase			)->						int:





						"""simple docstring"""



						return len(str(abs(__lowerCAmelCase			)			)			)


def __A					(   )->						None:





						"""simple docstring"""



						from collections.abc import Callable

						def benchmark_a_function(__lowerCAmelCase				,	__lowerCAmelCase			) -> None:
												_UpperCAmelCase             =    F"""{func.__name__}({value})"""
												_UpperCAmelCase             =    timeit(F"""__main__.{call}"""				,	setup='import __main__'			)
												print(F"""{call}: {func(__lowerCAmelCase			)} -- {timing} seconds"""			)

						for value in (262_144, 1_125_899_906_842_624, 1_267_650_600_228_229_401_496_703_205_376):
												for func in (num_digits, num_digits_fast, num_digits_faster):
																		benchmark_a_function(__lowerCAmelCase				,	__lowerCAmelCase			)
												print()


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		benchmark()
