

from __future__ import annotations

from collections.abc import Generator





def       __lowerCamelCase			(				):







							'''simple docstring'''






							snake_case_           =					{}
							snake_case_           =					2
							while True:
														snake_case_           =					factor_map.pop(UpperCamelCase__     ,							UpperCamelCase__   )
														if factor:
																					snake_case_           =					factor + prime
																					while x in factor_map:
																												x += factor
																					snake_case_           =					factor
														else:
																					snake_case_           =					prime
																					yield prime
														prime += 1





def       __lowerCamelCase			(				UpperCamelCase__ = 1E10   ):







							'''simple docstring'''






							snake_case_           =					sieve()

							snake_case_           =					1
							while True:
														snake_case_           =					next(UpperCamelCase__   )
														if (2 * prime * n) > limit:
																					return n
														# Ignore the next prime as the reminder will be 2.
														next(UpperCamelCase__   )
														n += 2


if __name__ == "__main__":
					print(solution())







from __future__ import annotations







class 				lowercase       :




				def __init__(      self   ,     snake_case ):
											snake_case_           =					TypeError(
											    'Matrices must be formed from a list of zero or more lists containing at '
											    'least one and the same number of values, each of which must be of type '
											    'int or float.' )
											if len(snake_case ) != 0:
																		snake_case_           =					len(rows[0] )
																		if cols == 0:
																									raise error
																		for row in rows:
																									if len(snake_case ) != cols:
																																raise error
																									for value in row:
																																if not isinstance(snake_case   ,     (int, float) ):
																																							raise error
																		snake_case_           =					rows
											else:
																		snake_case_           =					[]




				def 					a							(      self ):
											return [[row[i] for row in self.rows] for i in range(len(self.rows[0] ) )]




				@property
				def 					a							(      self ):
											return len(self.rows )




				@property
				def 					a							(      self ):
											return len(self.rows[0] )




				@property
				def 					a							(      self ):
											return (self.num_rows, self.num_columns)




				@property
				def 					a							(      self ):
											return self.order[0] == self.order[1]




				def 					a							(      self ):
											snake_case_           =					[
											    [0 if column_num != row_num else 1 for column_num in range(self.num_rows )]
											    for row_num in range(self.num_rows )
											]
											return Matrix(snake_case )




				def 					a							(      self ):
											if not self.is_square:
																		return 0
											if self.order == (0, 0):
																		return 1
											if self.order == (1, 1):
																		return int(self.rows[0][0] )
											if self.order == (2, 2):
																		return int(
																		    (self.rows[0][0] * self.rows[1][1])
																		    - (self.rows[0][1] * self.rows[1][0]) )
											else:
																		return sum(
																		    self.rows[0][column] * self.cofactors().rows[0][column]
																		    for column in range(self.num_columns ) )




				def 					a							(      self ):
											return bool(self.determinant() )




				def 					a							(      self   ,     snake_case   ,     snake_case ):
											snake_case_           =					[
											    [
											        self.rows[other_row][other_column]
											        for other_column in range(self.num_columns )
											        if other_column != column
											    ]
											    for other_row in range(self.num_rows )
											    if other_row != row
											]
											return Matrix(snake_case ).determinant()




				def 					a							(      self   ,     snake_case   ,     snake_case ):
											if (row + column) % 2 == 0:
																		return self.get_minor(snake_case   ,     snake_case )
											return -1 * self.get_minor(snake_case   ,     snake_case )




				def 					a							(      self ):
											return Matrix(
											    [
											        [self.get_minor(snake_case   ,     snake_case ) for column in range(self.num_columns )]
											        for row in range(self.num_rows )
											    ] )




				def 					a							(      self ):
											return Matrix(
											    [
											        [
											            self.minors().rows[row][column]
											            if (row + column) % 2 == 0
											            else self.minors().rows[row][column] * -1
											            for column in range(self.minors().num_columns )
											        ]
											        for row in range(self.minors().num_rows )
											    ] )




				def 					a							(      self ):
											snake_case_           =					[
											    [self.cofactors().rows[column][row] for column in range(self.num_columns )]
											    for row in range(self.num_rows )
											]
											return Matrix(snake_case )




				def 					a							(      self ):
											snake_case_           =					self.determinant()
											if not determinant:
																		raise TypeError('Only matrices with a non-zero determinant have an inverse' )
											return self.adjugate() * (1 / determinant)




				def __repr__(      self ):
											return str(self.rows )




				def __str__(      self ):
											if self.num_rows == 0:
																		return "[]"
											if self.num_rows == 1:
																		return "[[" + ". ".join(str(self.rows[0] ) ) + "]]"
											return (
											    "["
											    + "\n ".join(
											        [
											            '[' + '. '.join([str(snake_case ) for value in row] ) + '.]'
											            for row in self.rows
											        ] )
											    + "]"
											)




				def 					a							(      self   ,     snake_case   ,     snake_case = None ):
											snake_case_           =					TypeError('Row must be a list containing all ints and/or floats' )
											if not isinstance(snake_case   ,     snake_case ):
																		raise type_error
											for value in row:
																		if not isinstance(snake_case   ,     (int, float) ):
																									raise type_error
											if len(snake_case ) != self.num_columns:
																		raise ValueError(
																		    'Row must be equal in length to the other rows in the matrix' )
											if position is None:
																		self.rows.append(snake_case )
											else:
																		snake_case_           =					self.rows[0:position] + [row] + self.rows[position:]




				def 					a							(      self   ,     snake_case   ,     snake_case = None ):
											snake_case_           =					TypeError(
											    'Column must be a list containing all ints and/or floats' )
											if not isinstance(snake_case   ,     snake_case ):
																		raise type_error
											for value in column:
																		if not isinstance(snake_case   ,     (int, float) ):
																									raise type_error
											if len(snake_case ) != self.num_rows:
																		raise ValueError(
																		    'Column must be equal in length to the other columns in the matrix' )
											if position is None:
																		snake_case_           =					[self.rows[i] + [column[i]] for i in range(self.num_rows )]
											else:
																		snake_case_           =					[
																		    self.rows[i][0:position] + [column[i]] + self.rows[i][position:]
																		    for i in range(self.num_rows )
																		]




				def __eq__(      self   ,     snake_case ):
											if not isinstance(snake_case   ,     snake_case ):
																		return NotImplemented
											return self.rows == other.rows




				def __ne__(      self   ,     snake_case ):
											return not self == other




				def __neg__(      self ):
											return self * -1




				def __add__(      self   ,     snake_case ):
											if self.order != other.order:
																		raise ValueError('Addition requires matrices of the same order' )
											return Matrix(
											    [
											        [self.rows[i][j] + other.rows[i][j] for j in range(self.num_columns )]
											        for i in range(self.num_rows )
											    ] )




				def __sub__(      self   ,     snake_case ):
											if self.order != other.order:
																		raise ValueError('Subtraction requires matrices of the same order' )
											return Matrix(
											    [
											        [self.rows[i][j] - other.rows[i][j] for j in range(self.num_columns )]
											        for i in range(self.num_rows )
											    ] )




				def __mul__(      self   ,     snake_case ):
											if isinstance(snake_case   ,     (int, float) ):
																		return Matrix(
																		    [[int(element * other ) for element in row] for row in self.rows] )
											elif isinstance(snake_case   ,     snake_case ):
																		if self.num_columns != other.num_rows:
																									raise ValueError(
																									    'The number of columns in the first matrix must '
																									    'be equal to the number of rows in the second' )
																		return Matrix(
																		    [
																		        [Matrix.dot_product(snake_case   ,     snake_case ) for column in other.columns()]
																		        for row in self.rows
																		    ] )
											else:
																		raise TypeError(
																		    'A Matrix can only be multiplied by an int, float, or another matrix' )




				def __pow__(      self   ,     snake_case ):
											if not isinstance(snake_case   ,     snake_case ):
																		raise TypeError('A Matrix can only be raised to the power of an int' )
											if not self.is_square:
																		raise ValueError('Only square matrices can be raised to a power' )
											if other == 0:
																		return self.identity()
											if other < 0:
																		if self.is_invertable():
																									return self.inverse() ** (-other)
																		raise ValueError(
																		    'Only invertable matrices can be raised to a negative power' )
											snake_case_           =					self
											for _ in range(other - 1 ):
																		result *= self
											return result




				@classmethod
				def 					a							(      cls   ,     snake_case   ,     snake_case ):
											return sum(row[i] * column[i] for i in range(len(snake_case ) ) )


if __name__ == "__main__":
					import doctest

					doctest.testmod()




