import math


def      __lowerCAmelCase(		__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    Optional[Any]							=							math.loga(math.sqrt(4 * positive_integer + 1			) / 2 + 1 / 2			)

	return exponent == int(__UpperCAmelCase			)


def      __lowerCAmelCase(		__UpperCAmelCase = 1 / 12_345			):






	"""simple docstring"""





	_lowercase :    Dict							=							0
	_lowercase :    str							=							0

	_lowercase :    str							=							3
	while True:
		_lowercase :    Optional[int]							=							(integer**2 - 1) / 4
		# if candidate is an integer, then there is a partition for k
		if partition_candidate == int(__UpperCAmelCase			):
			_lowercase :    Union[str, Any]							=							int(__UpperCAmelCase			)
			total_partitions += 1
			if check_partition_perfect(__UpperCAmelCase			):
				perfect_partitions += 1
		if perfect_partitions > 0:
			if perfect_partitions / total_partitions < max_proportion:
				return int(__UpperCAmelCase			)
		integer += 1


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




"""simple docstring"""







def      __lowerCAmelCase(		__UpperCAmelCase     ,__UpperCAmelCase			):






	"""simple docstring"""





	_lowercase :    Any							=							len(__UpperCAmelCase			)
	_lowercase :    Any							=							len(__UpperCAmelCase			)
	_lowercase :    List[Any]							=							[[False for _ in range(m + 1			)] for _ in range(n + 1			)]
	_lowercase :    Union[str, Any]							=							True
	for i in range(__UpperCAmelCase			):
		for j in range(m + 1			):
			if dp[i][j]:
				if j < m and a[i].upper() == b[j]:
					_lowercase :    Optional[int]							=							True
				if a[i].islower():
					_lowercase :    Union[str, Any]							=							True
	return dp[n][m]


if __name__ == "__main__":
						import doctest

						doctest.testmod()



