






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			)








def 							SCREAMING_SNAKE_CASE_	(					UpperCAmelCase_	:     list				)			->						float:
       SCREAMING_SNAKE_CASE_							:				Dict		=0
       while len(UpperCAmelCase_				) > 1:
              SCREAMING_SNAKE_CASE_							:				Tuple		=0
              # Consider two files with minimum cost to be merged
              for _ in range(2				):
                     SCREAMING_SNAKE_CASE_							:				int		=files.index(min(UpperCAmelCase_				)				)
                     temp += files[min_index]
                     files.pop(UpperCAmelCase_				)
              files.append(UpperCAmelCase_				)
              optimal_merge_cost += temp
       return optimal_merge_cost


if __name__ == "__main__":
 import doctest

 doctest.testmod()
