


"""simple docstring"""







from __future__ import annotations

import math
import random
from collections.abc import Collection
from typing import overload




class 					UpperCamelCase_      :

	"""simple docstring"""





	def __init__(					self     :							List[Any]     ,     UpperCAmelCase__     :							Collection[float] | None = None  )			->      None:
						if components is None:
											__SCREAMING_SNAKE_CASE		  =							[]
						__SCREAMING_SNAKE_CASE		  =							list(__snake_case  )




	def __len__(					self     :							Union[str, Any]  )			->      int:
						return len(self.__components  )




	def __str__(					self     :							List[str]  )			->      str:
						return "(" + ",".join(map(__snake_case     ,     self.__components  )  ) + ")"




	def __add__(					self     :							List[str]     ,     UpperCAmelCase__     :							Vector  )			->      Vector:
						__SCREAMING_SNAKE_CASE		  =							len(self  )
						if size == len(__snake_case  ):
											__SCREAMING_SNAKE_CASE		  =							[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     :							Union[str, Any]     ,     UpperCAmelCase__     :							Vector  )			->      Vector:
						__SCREAMING_SNAKE_CASE		  =							len(self  )
						if size == len(__snake_case  ):
											__SCREAMING_SNAKE_CASE		  =							[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     :							Optional[int]     ,     UpperCAmelCase__     :							float  )			->      Vector:
						...




	@overload
	def __mul__(					self     :							List[Any]     ,     UpperCAmelCase__     :							Vector  )			->      float:
						...




	def __mul__(					self     :							Optional[Any]     ,     UpperCAmelCase__     :							float | Vector  )			->      float | Vector:
						if isinstance(__snake_case     ,     (float, int)  ):
											__SCREAMING_SNAKE_CASE		  =							[c * other for c in self.__components]
											return Vector(__snake_case  )
						elif isinstance(__snake_case     ,     __snake_case  ) and len(self  ) == len(__snake_case  ):
											__SCREAMING_SNAKE_CASE		  =							len(self  )
											__SCREAMING_SNAKE_CASE		  =							[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     UpperCAmelCase_ (					self     :							Any  )			->      Vector:
						return Vector(self.__components  )




	def     UpperCAmelCase_ (					self     :							Union[str, Any]     ,     UpperCAmelCase__     :							int  )			->      float:
						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     UpperCAmelCase_ (					self     :							Dict     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							float  )			->      None:
						assert -len(self.__components  ) <= pos < len(self.__components  )
						__SCREAMING_SNAKE_CASE		  =							value




	def     UpperCAmelCase_ (					self     :							List[Any]  )			->      float:
						if len(self.__components  ) == 0:
											raise Exception("Vector is empty"  )
						__SCREAMING_SNAKE_CASE		  =							[c**2 for c in self.__components]
						return math.sqrt(sum(__snake_case  )  )




	def     UpperCAmelCase_ (					self     :							List[str]     ,     UpperCAmelCase__     :							Vector     ,     UpperCAmelCase__     :							bool = False  )			->      float:
						__SCREAMING_SNAKE_CASE		  =							self * other
						__SCREAMING_SNAKE_CASE		  =							self.euclidean_length() * other.euclidean_length()
						if deg:
											return math.degrees(math.acos(num / den  )  )
						else:
											return math.acos(num / den  )




def   UpperCAmelCase__			(lowerCAmelCase_  ):




					'''simple docstring'''


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




def   UpperCAmelCase__			(lowerCAmelCase_    ,		lowerCAmelCase_  ):




					'''simple docstring'''


					assert isinstance(__lowerCAmelCase    ,		__lowerCAmelCase  ) and (isinstance(__lowerCAmelCase    ,		__lowerCAmelCase  ))
					__SCREAMING_SNAKE_CASE		  =							[0] * dimension
					__SCREAMING_SNAKE_CASE		  =							1
					return Vector(__lowerCAmelCase  )




def   UpperCAmelCase__			(lowerCAmelCase_    ,		lowerCAmelCase_    ,		lowerCAmelCase_  ):




					'''simple docstring'''


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




def   UpperCAmelCase__			(lowerCAmelCase_    ,		lowerCAmelCase_    ,		lowerCAmelCase_  ):




					'''simple docstring'''


					random.seed(__lowerCAmelCase  )
					__SCREAMING_SNAKE_CASE		  =							[random.randint(__lowerCAmelCase    ,		__lowerCAmelCase  ) for _ in range(__lowerCAmelCase  )]
					return Vector(__lowerCAmelCase  )




class 					UpperCamelCase_      :

	"""simple docstring"""





	def __init__(					self     :							int     ,     UpperCAmelCase__     :							list[list[float]]     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							int  )			->      None:
						__SCREAMING_SNAKE_CASE		  =							matrix
						__SCREAMING_SNAKE_CASE		  =							w
						__SCREAMING_SNAKE_CASE		  =							h




	def __str__(					self     :							int  )			->      str:
						__SCREAMING_SNAKE_CASE		  =							""
						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     :							int     ,     UpperCAmelCase__     :							Matrix  )			->      Matrix:
						if self.__width == other.width() and self.__height == other.height():
											__SCREAMING_SNAKE_CASE		  =							[]
											for i in range(self.__height  ):
																__SCREAMING_SNAKE_CASE		  =							[
																    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     :							List[Any]     ,     UpperCAmelCase__     :							Matrix  )			->      Matrix:
						if self.__width == other.width() and self.__height == other.height():
											__SCREAMING_SNAKE_CASE		  =							[]
											for i in range(self.__height  ):
																__SCREAMING_SNAKE_CASE		  =							[
																    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     :							int     ,     UpperCAmelCase__     :							float  )			->      Matrix:
						...




	@overload
	def __mul__(					self     :							str     ,     UpperCAmelCase__     :							Vector  )			->      Vector:
						...




	def __mul__(					self     :							Union[str, Any]     ,     UpperCAmelCase__     :							float | Vector  )			->      Vector | Matrix:
						if isinstance(__snake_case     ,     __snake_case  ):  # matrix-vector
											if len(__snake_case  ) == self.__width:
																__SCREAMING_SNAKE_CASE		  =							zero_vector(self.__height  )
																for i in range(self.__height  ):
																					__SCREAMING_SNAKE_CASE		  =							[
																					    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
											__SCREAMING_SNAKE_CASE		  =							[
											    [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     UpperCAmelCase_ (					self     :							str  )			->      int:
						return self.__height




	def     UpperCAmelCase_ (					self     :							Optional[Any]  )			->      int:
						return self.__width




	def     UpperCAmelCase_ (					self     :							List[str]     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							int  )			->      float:
						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     UpperCAmelCase_ (					self     :							Optional[Any]     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							float  )			->      None:
						if 0 <= x < self.__height and 0 <= y < self.__width:
											__SCREAMING_SNAKE_CASE		  =							value
						else:
											raise Exception("change_component: indices out of bounds"  )




	def     UpperCAmelCase_ (					self     :							Optional[Any]     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							int  )			->      float:
						if self.__height != self.__width:
											raise Exception("Matrix is not square"  )
						__SCREAMING_SNAKE_CASE		  =							self.__matrix[:x] + self.__matrix[x + 1 :]
						for i in range(len(__snake_case  )  ):
											__SCREAMING_SNAKE_CASE		  =							minor[i][:y] + minor[i][y + 1 :]
						return Matrix(__snake_case     ,     self.__width - 1     ,     self.__height - 1  ).determinant()




	def     UpperCAmelCase_ (					self     :							Optional[Any]     ,     UpperCAmelCase__     :							int     ,     UpperCAmelCase__     :							int  )			->      float:
						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     UpperCAmelCase_ (					self     :							Tuple  )			->      float:
						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:
											__SCREAMING_SNAKE_CASE		  =							[
											    self.__matrix[0][y] * self.cofactor(0     ,     __snake_case  ) for y in range(self.__width  )
											]
											return sum(__snake_case  )




def   UpperCAmelCase__			(lowerCAmelCase_  ):




					'''simple docstring'''


					__SCREAMING_SNAKE_CASE		  =							[[0] * n for _ in range(__lowerCAmelCase  )]
					return Matrix(__lowerCAmelCase    ,		__lowerCAmelCase    ,		__lowerCAmelCase  )




def   UpperCAmelCase__			(lowerCAmelCase_    ,		lowerCAmelCase_    ,		lowerCAmelCase_    ,		lowerCAmelCase_  ):




					'''simple docstring'''


					random.seed(__lowerCAmelCase  )
					__SCREAMING_SNAKE_CASE		  =							[
					    [random.randint(__lowerCAmelCase    ,		__lowerCAmelCase  ) for _ in range(__lowerCAmelCase  )] for _ in range(__lowerCAmelCase  )
					]
					return Matrix(__lowerCAmelCase    ,		__lowerCAmelCase    ,		__lowerCAmelCase  )







'''simple docstring'''




def 	__lowerCamelCase			(      __lowerCAmelCase       :							int		)		->  int:
   assert isinstance(__lowerCAmelCase      ,    __lowerCAmelCase		), F'''The input value of [n={number}] is not an integer'''

   if number == 1:
      return 2
   elif number < 1:
      snake_case 						=   F'''The input value of [n={number}] has to be > 0'''
      raise ValueError(__lowerCAmelCase		)
   else:
      snake_case 						=   sylvester(number - 1		)
      snake_case 						=   num - 1
      snake_case 						=   num
      return lower * upper + 1


if __name__ == "__main__":
 print(F"""The 8th number in Sylvester's sequence: {sylvester(8)}""")

