



from __future__ import annotations

from decimal import Decimal

from numpy import array


def    __lowerCamelCase   (  lowerCamelCase__		):

				"""simple docstring"""


				lowercase__	:       int						=	Decimal

				# Check if the provided matrix has 2 rows and 2 columns
				# since this implementation only works for 2x2 matrices
				if len(lowerCAmelCase__		) == 2 and len(matrix[0]		) == 2 and len(matrix[1]		) == 2:
								# Calculate the determinant of the matrix
								lowercase__	:       int						=	float(
								    d(matrix[0][0]		) * d(matrix[1][1]		) - d(matrix[1][0]		) * d(matrix[0][1]		)		)
								if determinant == 0:
												raise ValueError("This matrix has no inverse."		)

								# Creates a copy of the matrix with swapped positions of the elements
								lowercase__	:       Dict						=	[[0.0, 0.0], [0.0, 0.0]]
								lowercase__				,				lowercase__	:       Any						=	matrix[1][1], matrix[0][0]
								lowercase__				,				lowercase__	:       List[Any]						=	-matrix[1][0], -matrix[0][1]

								# Calculate the inverse of the matrix
								return [
								    [(float(d(lowerCAmelCase__		)		) / determinant) or 0.0 for n in row] for row in swapped_matrix
								]
				elif (
				    len(lowerCAmelCase__		) == 3
				    and len(matrix[0]		) == 3
				    and len(matrix[1]		) == 3
				    and len(matrix[2]		) == 3
				):
								# Calculate the determinant of the matrix using Sarrus rule
								lowercase__	:       str						=	float(
								    (
								        (d(matrix[0][0]		) * d(matrix[1][1]		) * d(matrix[2][2]		))
								        + (d(matrix[0][1]		) * d(matrix[1][2]		) * d(matrix[2][0]		))
								        + (d(matrix[0][2]		) * d(matrix[1][0]		) * d(matrix[2][1]		))
								    )
								    - (
								        (d(matrix[0][2]		) * d(matrix[1][1]		) * d(matrix[2][0]		))
								        + (d(matrix[0][1]		) * d(matrix[1][0]		) * d(matrix[2][2]		))
								        + (d(matrix[0][0]		) * d(matrix[1][2]		) * d(matrix[2][1]		))
								    )		)
								if determinant == 0:
												raise ValueError("This matrix has no inverse."		)

								# Creating cofactor matrix
								lowercase__	:       int						=	[
								    [d(0.0		), d(0.0		), d(0.0		)],
								    [d(0.0		), d(0.0		), d(0.0		)],
								    [d(0.0		), d(0.0		), d(0.0		)],
								]
								lowercase__	:       Tuple						=	(d(matrix[1][1]		) * d(matrix[2][2]		)) - (
								    d(matrix[1][2]		) * d(matrix[2][1]		)
								)
								lowercase__	:       str						=	-(
								    (d(matrix[1][0]		) * d(matrix[2][2]		)) - (d(matrix[1][2]		) * d(matrix[2][0]		))
								)
								lowercase__	:       str						=	(d(matrix[1][0]		) * d(matrix[2][1]		)) - (
								    d(matrix[1][1]		) * d(matrix[2][0]		)
								)
								lowercase__	:       str						=	-(
								    (d(matrix[0][1]		) * d(matrix[2][2]		)) - (d(matrix[0][2]		) * d(matrix[2][1]		))
								)
								lowercase__	:       List[str]						=	(d(matrix[0][0]		) * d(matrix[2][2]		)) - (
								    d(matrix[0][2]		) * d(matrix[2][0]		)
								)
								lowercase__	:       Optional[Any]						=	-(
								    (d(matrix[0][0]		) * d(matrix[2][1]		)) - (d(matrix[0][1]		) * d(matrix[2][0]		))
								)
								lowercase__	:       Tuple						=	(d(matrix[0][1]		) * d(matrix[1][2]		)) - (
								    d(matrix[0][2]		) * d(matrix[1][1]		)
								)
								lowercase__	:       List[Any]						=	-(
								    (d(matrix[0][0]		) * d(matrix[1][2]		)) - (d(matrix[0][2]		) * d(matrix[1][0]		))
								)
								lowercase__	:       Dict						=	(d(matrix[0][0]		) * d(matrix[1][1]		)) - (
								    d(matrix[0][1]		) * d(matrix[1][0]		)
								)

								# Transpose the cofactor matrix (Adjoint matrix)
								lowercase__	:       Any						=	array(lowerCAmelCase__		)
								for i in range(3		):
												for j in range(3		):
																lowercase__	:       Optional[int]						=	cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
								lowercase__	:       List[Any]						=	array(lowerCAmelCase__		)
								for i in range(3		):
												for j in range(3		):
																inverse_matrix[i][j] /= d(lowerCAmelCase__		)

        # Calculate the inverse of the matrix
								return [[float(d(lowerCAmelCase__		)		) or 0.0 for n in row] for row in inverse_matrix]
				raise ValueError("Please provide a matrix of size 2x2 or 3x3."		)





"""simple docstring"""






import math


def      UpperCAmelCase__			(							lowerCAmelCase__	:int						)					->       bool:
      '''simple docstring'''


      lowercase								=      math.loga(math.sqrt(4 * positive_integer + 1						) / 2 + 1 / 2						)

      return exponent == int(lowerCAmelCase__						)


def      UpperCAmelCase__			(							lowerCAmelCase__	:float = 1 / 1_2_3_4_5						)					->       int:
      '''simple docstring'''


      lowercase								=      0
      lowercase								=      0

      lowercase								=      3
      while True:
            lowercase								=      (integer**2 - 1) / 4
            # if candidate is an integer, then there is a partition for k
            if partition_candidate == int(lowerCAmelCase__						):
                  lowercase								=      int(lowerCAmelCase__						)
                  total_partitions += 1
                  if check_partition_perfect(lowerCAmelCase__						):
                        perfect_partitions += 1
            if perfect_partitions > 0:
                  if perfect_partitions / total_partitions < max_proportion:
                        return int(lowerCAmelCase__						)
            integer += 1


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



