






from __future__ import annotations

import math
import random
from collections.abc import Collection
from typing import overload
class     lowerCamelCase_					:







				def __init__(    self							,       __lowerCAmelCase = None  ):
								"""simple docstring"""
								if components is None:
												__magic_name__		:List[Any]   =   []
								__magic_name__		:str   =   list(__lowerCAmelCase  )







				def __len__(    self  ):
								"""simple docstring"""
								return len(self.__components  )







				def __str__(    self  ):
								"""simple docstring"""
								return "(" + ",".join(map(__lowerCAmelCase							,       self.__components  )  ) + ")"







				def __add__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								__magic_name__		:Dict   =   len(self  )
								if size == len(__lowerCAmelCase  ):
												__magic_name__		:Dict   =   [self.__components[i] + other.component(__lowerCAmelCase  ) for i in range(__lowerCAmelCase  )]
												return Vector(__lowerCAmelCase  )
								else:
												raise Exception('''must have the same size'''  )







				def __sub__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								__magic_name__		:Union[str, Any]   =   len(self  )
								if size == len(__lowerCAmelCase  ):
												__magic_name__		:Tuple   =   [self.__components[i] - other.component(__lowerCAmelCase  ) for i in range(__lowerCAmelCase  )]
												return Vector(__lowerCAmelCase  )
								else:  # error case
												raise Exception('''must have the same size'''  )







				@overload
				def __mul__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								...







				@overload
				def __mul__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								...







				def __mul__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if isinstance(__lowerCAmelCase							,       (float, int)  ):
												__magic_name__		:List[str]   =   [c * other for c in self.__components]
												return Vector(__lowerCAmelCase  )
								elif isinstance(__lowerCAmelCase							,       __lowerCAmelCase  ) and len(self  ) == len(__lowerCAmelCase  ):
												__magic_name__		:Optional[int]   =   len(self  )
												__magic_name__		:Union[str, Any]   =   [self.__components[i] * other.component(__lowerCAmelCase  ) for i in range(__lowerCAmelCase  )]
												return sum(__lowerCAmelCase  )
								else:  # error case
												raise Exception('''invalid operand!'''  )







				def       A						(    self  ):
								"""simple docstring"""
								return Vector(self.__components  )







				def       A						(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if isinstance(__lowerCAmelCase							,       __lowerCAmelCase  ) and -len(self.__components  ) <= i < len(self.__components  ):
												return self.__components[i]
								else:
												raise Exception('''index out of range'''  )







				def       A						(    self							,       __lowerCAmelCase							,       __lowerCAmelCase  ):
								"""simple docstring"""
								assert -len(self.__components  ) <= pos < len(self.__components  )
								__magic_name__		:Optional[int]   =   value







				def       A						(    self  ):
								"""simple docstring"""
								if len(self.__components  ) == 0:
												raise Exception('''Vector is empty'''  )
								__magic_name__		:Dict   =   [c**2 for c in self.__components]
								return math.sqrt(sum(__lowerCAmelCase  )  )







				def       A						(    self							,       __lowerCAmelCase							,       __lowerCAmelCase = False  ):
								"""simple docstring"""
								__magic_name__		:str   =   self * other
								__magic_name__		:int   =   self.euclidean_length() * other.euclidean_length()
								if deg:
												return math.degrees(math.acos(num / den  )  )
								else:
												return math.acos(num / den  )







def   __lowercase       (			snake_case			):







				"""simple docstring"""






				assert isinstance(snake_case,	snake_case			)
				return Vector([0] * dimension			)

def   __lowercase       (			snake_case,	snake_case			):







				"""simple docstring"""






				assert isinstance(snake_case,	snake_case			) and (isinstance(snake_case,	snake_case			))
				__magic_name__		:List[str]   =   [0] * dimension
				__magic_name__		:int   =   1
				return Vector(snake_case			)

def   __lowercase       (			snake_case,	snake_case,	snake_case			):







				"""simple docstring"""






				assert (
				    isinstance(snake_case,	snake_case			)
				    and isinstance(snake_case,	snake_case			)
				    and (isinstance(snake_case,	(int, float)			))
				)
				return x * scalar + y

def   __lowercase       (			snake_case,	snake_case,	snake_case			):







				"""simple docstring"""






				random.seed(snake_case			)
				__magic_name__		:Any   =   [random.randint(snake_case,	snake_case			) for _ in range(snake_case			)]
				return Vector(snake_case			)
class     lowerCamelCase_					:







				def __init__(    self							,       __lowerCAmelCase							,       __lowerCAmelCase							,       __lowerCAmelCase  ):
								"""simple docstring"""
								__magic_name__		:Optional[Any]   =   matrix
								__magic_name__		:Union[str, Any]   =   w
								__magic_name__		:Union[str, Any]   =   h







				def __str__(    self  ):
								"""simple docstring"""
								__magic_name__		:List[Any]   =   ''''''
								for i in range(self.__height  ):
												ans += "|"
												for j in range(self.__width  ):
																if j < self.__width - 1:
																				ans += str(self.__matrix[i][j]  ) + ","
																else:
																				ans += str(self.__matrix[i][j]  ) + "|\n"
								return ans







				def __add__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if self.__width == other.width() and self.__height == other.height():
												__magic_name__		:str   =   []
												for i in range(self.__height  ):
																__magic_name__		:List[str]   =   [
																    self.__matrix[i][j] + other.component(__lowerCAmelCase							,       __lowerCAmelCase  )
																    for j in range(self.__width  )
																]
																matrix.append(__lowerCAmelCase  )
												return Matrix(__lowerCAmelCase							,       self.__width							,       self.__height  )
								else:
												raise Exception('''matrix must have the same dimension!'''  )







				def __sub__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if self.__width == other.width() and self.__height == other.height():
												__magic_name__		:int   =   []
												for i in range(self.__height  ):
																__magic_name__		:Tuple   =   [
																    self.__matrix[i][j] - other.component(__lowerCAmelCase							,       __lowerCAmelCase  )
																    for j in range(self.__width  )
																]
																matrix.append(__lowerCAmelCase  )
												return Matrix(__lowerCAmelCase							,       self.__width							,       self.__height  )
								else:
												raise Exception('''matrices must have the same dimension!'''  )







				@overload
				def __mul__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								...







				@overload
				def __mul__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								...







				def __mul__(    self							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if isinstance(__lowerCAmelCase							,       __lowerCAmelCase  ):  # matrix-vector
												if len(__lowerCAmelCase  ) == self.__width:
																__magic_name__		:Tuple   =   zero_vector(self.__height  )
																for i in range(self.__height  ):
																				__magic_name__		:Optional[int]   =   [
																				    self.__matrix[i][j] * other.component(__lowerCAmelCase  )
																				    for j in range(self.__width  )
																				]
																				ans.change_component(__lowerCAmelCase							,       sum(__lowerCAmelCase  )  )
																return ans
												else:
																raise Exception(
																    '''vector must have the same size as the '''
																    '''number of columns of the matrix!'''  )
								elif isinstance(__lowerCAmelCase							,       (int, float)  ):  # matrix-scalar
												__magic_name__		:Optional[Any]   =   [
												    [self.__matrix[i][j] * other for j in range(self.__width  )]
												    for i in range(self.__height  )
												]
												return Matrix(__lowerCAmelCase							,       self.__width							,       self.__height  )
								return None







				def       A						(    self  ):
								"""simple docstring"""
								return self.__height







				def       A						(    self  ):
								"""simple docstring"""
								return self.__width







				def       A						(    self							,       __lowerCAmelCase							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if 0 <= x < self.__height and 0 <= y < self.__width:
												return self.__matrix[x][y]
								else:
												raise Exception('''change_component: indices out of bounds'''  )







				def       A						(    self							,       __lowerCAmelCase							,       __lowerCAmelCase							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if 0 <= x < self.__height and 0 <= y < self.__width:
												__magic_name__		:Union[str, Any]   =   value
								else:
												raise Exception('''change_component: indices out of bounds'''  )







				def       A						(    self							,       __lowerCAmelCase							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if self.__height != self.__width:
												raise Exception('''Matrix is not square'''  )
								__magic_name__		:Optional[int]   =   self.__matrix[:x] + self.__matrix[x + 1 :]
								for i in range(len(__lowerCAmelCase  )  ):
												__magic_name__		:Optional[Any]   =   minor[i][:y] + minor[i][y + 1 :]
								return Matrix(__lowerCAmelCase							,       self.__width - 1							,       self.__height - 1  ).determinant()







				def       A						(    self							,       __lowerCAmelCase							,       __lowerCAmelCase  ):
								"""simple docstring"""
								if self.__height != self.__width:
												raise Exception('''Matrix is not square'''  )
								if 0 <= x < self.__height and 0 <= y < self.__width:
												return (-1) ** (x + y) * self.minor(__lowerCAmelCase							,       __lowerCAmelCase  )
								else:
												raise Exception('''Indices out of bounds'''  )







				def       A						(    self  ):
								"""simple docstring"""
								if self.__height != self.__width:
												raise Exception('''Matrix is not square'''  )
								if self.__height < 1:
												raise Exception('''Matrix has no element'''  )
								elif self.__height == 1:
												return self.__matrix[0][0]
								elif self.__height == 2:
												return (
												    self.__matrix[0][0] * self.__matrix[1][1]
												    - self.__matrix[0][1] * self.__matrix[1][0]
												)
								else:
												__magic_name__		:int   =   [
												    self.__matrix[0][y] * self.cofactor(0							,       __lowerCAmelCase  ) for y in range(self.__width  )
												]
												return sum(__lowerCAmelCase  )







def   __lowercase       (			snake_case			):







				"""simple docstring"""






				__magic_name__		:list[list[float]]   =   [[0] * n for _ in range(snake_case			)]
				return Matrix(snake_case,	snake_case,	snake_case			)

def   __lowercase       (			snake_case,	snake_case,	snake_case,	snake_case			):







				"""simple docstring"""






				random.seed(snake_case			)
				__magic_name__		:list[list[float]]   =   [
				    [random.randint(snake_case,	snake_case			) for _ in range(snake_case			)] for _ in range(snake_case			)
				]
				return Matrix(snake_case,	snake_case,	snake_case			)










snake_case__				:	List[Any]					       =					[
    999,
    800,
    799,
    600,
    599,
    500,
    400,
    399,
    377,
    355,
    333,
    311,
    288,
    266,
    244,
    222,
    200,
    199,
    177,
    155,
    133,
    111,
    88,
    66,
    44,
    22,
    0,
]

snake_case__				:	Tuple					       =					[
    999,
    976,
    952,
    928,
    905,
    882,
    858,
    857,
    810,
    762,
    715,
    714,
    572,
    429,
    428,
    286,
    285,
    238,
    190,
    143,
    142,
    118,
    95,
    71,
    47,
    24,
    0,
]

snake_case__				:	List[Any]					       =					[
    999,
    988,
    977,
    966,
    955,
    944,
    933,
    922,
    911,
    900,
    899,
    879,
    859,
    840,
    820,
    800,
    799,
    766,
    733,
    700,
    699,
    650,
    600,
    599,
    500,
    499,
    400,
    399,
    350,
    300,
    299,
    266,
    233,
    200,
    199,
    179,
    159,
    140,
    120,
    100,
    99,
    88,
    77,
    66,
    55,
    44,
    33,
    22,
    11,
    0,
]

snake_case__				:	List[str]					       =					[
    999,
    995,
    992,
    989,
    985,
    981,
    978,
    975,
    971,
    967,
    964,
    961,
    957,
    956,
    951,
    947,
    942,
    937,
    933,
    928,
    923,
    919,
    914,
    913,
    908,
    903,
    897,
    892,
    887,
    881,
    876,
    871,
    870,
    864,
    858,
    852,
    846,
    840,
    834,
    828,
    827,
    820,
    813,
    806,
    799,
    792,
    785,
    784,
    777,
    770,
    763,
    756,
    749,
    742,
    741,
    733,
    724,
    716,
    707,
    699,
    698,
    688,
    677,
    666,
    656,
    655,
    645,
    634,
    623,
    613,
    612,
    598,
    584,
    570,
    569,
    555,
    541,
    527,
    526,
    505,
    484,
    483,
    462,
    440,
    439,
    396,
    395,
    352,
    351,
    308,
    307,
    264,
    263,
    220,
    219,
    176,
    132,
    88,
    44,
    0,
]

snake_case__				:	Any					       =					[
    999,
    997,
    995,
    992,
    990,
    988,
    986,
    984,
    981,
    979,
    977,
    975,
    972,
    970,
    968,
    966,
    964,
    961,
    959,
    957,
    956,
    954,
    951,
    949,
    946,
    944,
    941,
    939,
    936,
    934,
    931,
    929,
    926,
    924,
    921,
    919,
    916,
    914,
    913,
    910,
    907,
    905,
    902,
    899,
    896,
    893,
    891,
    888,
    885,
    882,
    879,
    877,
    874,
    871,
    870,
    867,
    864,
    861,
    858,
    855,
    852,
    849,
    846,
    843,
    840,
    837,
    834,
    831,
    828,
    827,
    824,
    821,
    817,
    814,
    811,
    808,
    804,
    801,
    798,
    795,
    791,
    788,
    785,
    784,
    780,
    777,
    774,
    770,
    766,
    763,
    760,
    756,
    752,
    749,
    746,
    742,
    741,
    737,
    733,
    730,
    726,
    722,
    718,
    714,
    710,
    707,
    703,
    699,
    698,
    694,
    690,
    685,
    681,
    677,
    673,
    669,
    664,
    660,
    656,
    655,
    650,
    646,
    641,
    636,
    632,
    627,
    622,
    618,
    613,
    612,
    607,
    602,
    596,
    591,
    586,
    580,
    575,
    570,
    569,
    563,
    557,
    551,
    545,
    539,
    533,
    527,
    526,
    519,
    512,
    505,
    498,
    491,
    484,
    483,
    474,
    466,
    457,
    449,
    440,
    439,
    428,
    418,
    407,
    396,
    395,
    381,
    366,
    352,
    351,
    330,
    308,
    307,
    286,
    264,
    263,
    242,
    220,
    219,
    176,
    175,
    132,
    131,
    88,
    44,
    0,
]

snake_case__				:	int					       =					[
    999,
    991,
    982,
    974,
    966,
    958,
    950,
    941,
    933,
    925,
    916,
    908,
    900,
    899,
    874,
    850,
    825,
    800,
    799,
    700,
    600,
    500,
    400,
    300,
    200,
    100,
    0,
]

snake_case__				:	Any					       =					[
    999,
    992,
    985,
    978,
    971,
    964,
    957,
    949,
    942,
    935,
    928,
    921,
    914,
    907,
    900,
    899,
    879,
    859,
    840,
    820,
    800,
    799,
    766,
    733,
    700,
    699,
    650,
    600,
    599,
    500,
    499,
    400,
    399,
    300,
    299,
    200,
    199,
    100,
    99,
    0,
]

snake_case__				:	Union[str, Any]					       =					[
    999,
    996,
    992,
    989,
    985,
    982,
    979,
    975,
    972,
    968,
    965,
    961,
    958,
    955,
    951,
    948,
    944,
    941,
    938,
    934,
    931,
    927,
    924,
    920,
    917,
    914,
    910,
    907,
    903,
    900,
    899,
    891,
    884,
    876,
    869,
    861,
    853,
    846,
    838,
    830,
    823,
    815,
    808,
    800,
    799,
    788,
    777,
    766,
    755,
    744,
    733,
    722,
    711,
    700,
    699,
    688,
    677,
    666,
    655,
    644,
    633,
    622,
    611,
    600,
    599,
    585,
    571,
    557,
    542,
    528,
    514,
    500,
    499,
    485,
    471,
    457,
    442,
    428,
    414,
    400,
    399,
    379,
    359,
    340,
    320,
    300,
    299,
    279,
    259,
    240,
    220,
    200,
    199,
    166,
    133,
    100,
    99,
    66,
    33,
    0,
]





