'''simple docstring'''





from __future__ import annotations

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







							def __init__(				self     :    List[str]	,					__snake_case     :    Collection[float] | None = None		):
										if components is None:
													lowerCamelCase						:Optional[Any]						  =     []
										lowerCamelCase						:Dict						  =     list(__snake_case		)







							def __len__(				self     :    Dict		):
										return len(self.__components		)







							def __str__(				self     :    Union[str, Any]		):
										return "(" + ",".join(map(__snake_case	,					self.__components		)		) + ")"







							def __add__(				self     :    Optional[Any]	,					__snake_case     :    Vector		):
										lowerCamelCase						:Any						  =     len(self		)
										if size == len(__snake_case		):
													lowerCamelCase						:Tuple						  =     [self.__components[i] + other.component(__snake_case		) for i in range(__snake_case		)]
													return Vector(__snake_case		)
										else:
													raise Exception('''must have the same size'''		)







							def __sub__(				self     :    Any	,					__snake_case     :    Vector		):
										lowerCamelCase						:Dict						  =     len(self		)
										if size == len(__snake_case		):
													lowerCamelCase						:List[str]						  =     [self.__components[i] - other.component(__snake_case		) for i in range(__snake_case		)]
													return Vector(__snake_case		)
										else:  # error case
													raise Exception('''must have the same size'''		)







							@overload
							def __mul__(				self     :    Any	,					__snake_case     :    float		):
										...







							@overload
							def __mul__(				self     :    Optional[int]	,					__snake_case     :    Vector		):
										...







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







							def 				snake_case  (				self     :    Optional[int]		):
										return Vector(self.__components		)







							def 				snake_case  (				self     :    Dict	,					__snake_case     :    int		):
										if isinstance(__snake_case	,					__snake_case		) and -len(self.__components		) <= i < len(self.__components		):
													return self.__components[i]
										else:
													raise Exception('''index out of range'''		)







							def 				snake_case  (				self     :    Union[str, Any]	,					__snake_case     :    int	,					__snake_case     :    float		):
										assert -len(self.__components		) <= pos < len(self.__components		)
										lowerCamelCase						:Any						  =     value







							def 				snake_case  (				self     :    str		):
										if len(self.__components		) == 0:
													raise Exception('''Vector is empty'''		)
										lowerCamelCase						:Optional[Any]						  =     [c**2 for c in self.__components]
										return math.sqrt(sum(__snake_case		)		)







							def 				snake_case  (				self     :    Dict	,					__snake_case     :    Vector	,					__snake_case     :    bool = False		):
										lowerCamelCase						:Dict						  =     self * other
										lowerCamelCase						:Any						  =     self.euclidean_length() * other.euclidean_length()
										if deg:
													return math.degrees(math.acos(num / den		)		)
										else:
													return math.acos(num / den		)





def  _lowerCamelCase					(   a_				: int):
			assert isinstance(_lowerCamelCase						,			_lowerCamelCase)
			return Vector([0] * dimension)





def  _lowerCamelCase					(   a_				: int						,			a_				: int):
			assert isinstance(_lowerCamelCase						,			_lowerCamelCase) and (isinstance(_lowerCamelCase						,			_lowerCamelCase))
			lowerCamelCase						:str						  =     [0] * dimension
			lowerCamelCase						:Union[str, Any]						  =     1
			return Vector(_lowerCamelCase)





def  _lowerCamelCase					(   a_				: float						,			a_				: Vector						,			a_				: Vector):
			assert (
			    isinstance(_lowerCamelCase						,			_lowerCamelCase)
			    and isinstance(_lowerCamelCase						,			_lowerCamelCase)
			    and (isinstance(_lowerCamelCase						,			(int, float)))
			)
			return x * scalar + y





def  _lowerCamelCase					(   a_				: int						,			a_				: int						,			a_				: int):
			random.seed(_lowerCamelCase)
			lowerCamelCase						:Optional[Any]						  =     [random.randint(_lowerCamelCase						,			_lowerCamelCase) for _ in range(_lowerCamelCase)]
			return Vector(_lowerCamelCase)
class   _lowerCAmelCase		:







							def __init__(				self     :    Optional[Any]	,					__snake_case     :    list[list[float]]	,					__snake_case     :    int	,					__snake_case     :    int		):
										lowerCamelCase						:Optional[int]						  =     matrix
										lowerCamelCase						:Dict						  =     w
										lowerCamelCase						:int						  =     h







							def __str__(				self     :    int		):
										lowerCamelCase						:Optional[int]						  =     ""
										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     :    Optional[Any]	,					__snake_case     :    Matrix		):
										if self.__width == other.width() and self.__height == other.height():
													lowerCamelCase						:Union[str, Any]						  =     []
													for i in range(self.__height		):
																lowerCamelCase						:Any						  =     [
																    self.__matrix[i][j] + other.component(__snake_case	,					__snake_case		)
																    for j in range(self.__width		)
																]
																matrix.append(__snake_case		)
													return Matrix(__snake_case	,					self.__width	,					self.__height		)
										else:
													raise Exception('''matrix must have the same dimension!'''		)







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







							@overload
							def __mul__(				self     :    Tuple	,					__snake_case     :    float		):
										...







							@overload
							def __mul__(				self     :    int	,					__snake_case     :    Vector		):
										...







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







							def 				snake_case  (				self     :    List[str]		):
										return self.__height







							def 				snake_case  (				self     :    int		):
										return self.__width







							def 				snake_case  (				self     :    str	,					__snake_case     :    int	,					__snake_case     :    int		):
										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 				snake_case  (				self     :    Any	,					__snake_case     :    int	,					__snake_case     :    int	,					__snake_case     :    float		):
										if 0 <= x < self.__height and 0 <= y < self.__width:
													lowerCamelCase						:Dict						  =     value
										else:
													raise Exception('''change_component: indices out of bounds'''		)







							def 				snake_case  (				self     :    Optional[Any]	,					__snake_case     :    int	,					__snake_case     :    int		):
										if self.__height != self.__width:
													raise Exception('''Matrix is not square'''		)
										lowerCamelCase						:Union[str, Any]						  =     self.__matrix[:x] + self.__matrix[x + 1 :]
										for i in range(len(__snake_case		)		):
													lowerCamelCase						:Any						  =     minor[i][:y] + minor[i][y + 1 :]
										return Matrix(__snake_case	,					self.__width - 1	,					self.__height - 1		).determinant()







							def 				snake_case  (				self     :    Dict	,					__snake_case     :    int	,					__snake_case     :    int		):
										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(__snake_case	,					__snake_case		)
										else:
													raise Exception('''Indices out of bounds'''		)







							def 				snake_case  (				self     :    List[str]		):
										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:
													lowerCamelCase						:Dict						  =     [
													    self.__matrix[0][y] * self.cofactor(0	,					__snake_case		) for y in range(self.__width		)
													]
													return sum(__snake_case		)





def  _lowerCamelCase					(   a_				: int):
			lowerCamelCase						:list[list[float]]						  =     [[0] * n for _ in range(_lowerCamelCase)]
			return Matrix(_lowerCamelCase						,			_lowerCamelCase						,			_lowerCamelCase)





def  _lowerCamelCase					(   a_				: int						,			a_				: int						,			a_				: int						,			a_				: int):
			random.seed(_lowerCamelCase)
			lowerCamelCase						:list[list[float]]						  =     [
			    [random.randint(_lowerCamelCase						,			_lowerCamelCase) for _ in range(_lowerCamelCase)] for _ in range(_lowerCamelCase)
			]
			return Matrix(_lowerCamelCase						,			_lowerCamelCase						,			_lowerCamelCase)




def  _lowerCamelCase					(   a_				: list):

			if not grid or not grid[0]:
						raise TypeError('''The grid does not contain the appropriate information''')

			for cell_n in range(1						,			len(grid[0])):
						grid[0][cell_n] += grid[0][cell_n - 1]
			lowerCamelCase						:Any						  =     grid[0]

			for row_n in range(1						,			len(a_)):
						lowerCamelCase						:List[str]						  =     grid[row_n]
						lowerCamelCase						:Union[str, Any]						  =     fill_row(a_						,			a_)
						lowerCamelCase						:List[Any]						  =     grid[row_n]

			return grid[-1][-1]





def  _lowerCamelCase					(   a_				: list						,			a_				: list):
			current_row[0] += row_above[0]
			for cell_n in range(1						,			len(a_)):
						current_row[cell_n] += min(current_row[cell_n - 1]						,			row_above[cell_n])

			return current_row


if __name__ == "__main__":
			import doctest

			doctest.testmod()



