






'''simple docstring'''







import mpmath  # for roots of unity
import numpy as np





class 	SCREAMING_SNAKE_CASE					:






			def __init__( self      :			int  ,						__SCREAMING_SNAKE_CASE      :			Union[str, Any]=None  ,						__SCREAMING_SNAKE_CASE      :			Optional[int]=None					)			->			List[str]:
							# Input as list
							a_		:  Tuple											=      list(poly_a or [0]					)[:]
							a_		:  int											=      list(poly_b or [0]					)[:]

							# Remove leading zero coefficients
							while self.polyA[-1] == 0:
											self.polyA.pop()
							a_		:  str											=      len(self.polyA					)

							while self.polyB[-1] == 0:
											self.polyB.pop()
							a_		:  Tuple											=      len(self.polyB					)

							# Add 0 to make lengths equal a power of 2
							a_		:  Tuple											=      int(
							    2 ** np.ceil(np.loga(len(self.polyA					) + len(self.polyB					) - 1					)					)					)

							while len(self.polyA					) < self.c_max_length:
											self.polyA.append(0					)
							while len(self.polyB					) < self.c_max_length:
											self.polyB.append(0					)
							# A complex root used for the fourier transform
							a_		:  Optional[Any]											=      complex(mpmath.root(x=1  ,						n=self.c_max_length  ,						k=1					)					)

							# The product
							a_		:  int											=      self.__multiply()






			def 			SCREAMING_SNAKE_CASE      ( self      :			Optional[Any]  ,						__SCREAMING_SNAKE_CASE      :			Dict					)			->			List[str]:
							a_		:  Tuple											=      [[x] for x in self.polyA] if which == '''A''' else [[x] for x in self.polyB]
							# Corner case
							if len(__SCREAMING_SNAKE_CASE					) <= 1:
											return dft[0]
							#
							a_		:  Union[str, Any]											=      self.c_max_length // 2
							while next_ncol > 0:
											a_		:  List[str]											=      [[] for i in range(__SCREAMING_SNAKE_CASE					)]
											a_		:  Optional[Any]											=      self.root**next_ncol

											# First half of next step
											a_		:  Optional[int]											=      1
											for j in range(self.c_max_length // (next_ncol * 2)					):
															for i in range(__SCREAMING_SNAKE_CASE					):
																			new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]					)
															current_root *= root
											# Second half of next step
											a_		:  List[str]											=      1
											for j in range(self.c_max_length // (next_ncol * 2)					):
															for i in range(__SCREAMING_SNAKE_CASE					):
																			new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]					)
															current_root *= root
											# Update
											a_		:  Any											=      new_dft
											a_		:  str											=      next_ncol // 2
							return dft[0]






			def 			SCREAMING_SNAKE_CASE      ( self      :			Union[str, Any]					)			->			Tuple:
							a_		:  List[Any]											=      self.__dft('''A'''					)
							a_		:  Dict											=      self.__dft('''B'''					)
							a_		:  Union[str, Any]											=      [[dft_a[i] * dft_b[i] for i in range(self.c_max_length					)]]
							del dft_a
							del dft_b

							# Corner Case
							if len(inverce_c[0]					) <= 1:
											return inverce_c[0]
							# Inverse DFT
							a_		:  List[Any]											=      2
							while next_ncol <= self.c_max_length:
											a_		:  int											=      [[] for i in range(__SCREAMING_SNAKE_CASE					)]
											a_		:  Tuple											=      self.root ** (next_ncol // 2)
											a_		:  Dict											=      1
											# First half of next step
											for j in range(self.c_max_length // next_ncol					):
															for i in range(next_ncol // 2					):
																			# Even positions
																			new_inverse_c[i].append(
																			    (
																			        inverce_c[i][j]
																			        + inverce_c[i][j + self.c_max_length // next_ncol]
																			    )
																			    / 2					)
																			# Odd positions
																			new_inverse_c[i + next_ncol // 2].append(
																			    (
																			        inverce_c[i][j]
																			        - inverce_c[i][j + self.c_max_length // next_ncol]
																			    )
																			    / (2 * current_root)					)
															current_root *= root
											# Update
											a_		:  Optional[Any]											=      new_inverse_c
											next_ncol *= 2
							# Unpack
							a_		:  Optional[int]											=      [round(x[0].real  ,						8					) + round(x[0].imag  ,						8					) * 1J for x in inverce_c]

							# Remove leading 0's
							while inverce_c[-1] == 0:
											inverce_c.pop()
							return inverce_c







			def __str__( self      :			Optional[int]					)			->			List[Any]:
							a_		:  Any											=      '''A = ''' + ''' + '''.join(
							    f'{coef}*x^{i}' for coef, i in enumerate(self.polyA[: self.len_A]					)					)
							a_		:  Any											=      '''B = ''' + ''' + '''.join(
							    f'{coef}*x^{i}' for coef, i in enumerate(self.polyB[: self.len_B]					)					)
							a_		:  Optional[Any]											=      '''A*B = ''' + ''' + '''.join(
							    f'{coef}*x^{i}' for coef, i in enumerate(self.product					)					)

							return f'{a}\n{b}\n{c}'


# Unit tests
if __name__ == "__main__":
		import doctest

		doctest.testmod()












'''simple docstring'''







from __future__ import annotations





def 		_UpperCAmelCase   (							__A							:   tuple[int, int]						,		__A							:   int      ):
				a_				,			a_		:  List[str]											=      position
				a_		:  Optional[int]											=      [
				    (y + 1, x + 2),
				    (y - 1, x + 2),
				    (y + 1, x - 2),
				    (y - 1, x - 2),
				    (y + 2, x + 1),
				    (y + 2, x - 1),
				    (y - 2, x + 1),
				    (y - 2, x - 1),
				]
				a_		:  Any											=      []

				for position in positions:
								a_				,			a_		:  Dict											=      position
								if 0 <= y_test < n and 0 <= x_test < n:
												permissible_positions.append(__A      )

				return permissible_positions





def 		_UpperCAmelCase   (							__A							:   list[list[int]]      ):
				return not any(elem == 0 for row in board for elem in row      )





def 		_UpperCAmelCase   (							__A							:   list[list[int]]						,		__A							:   tuple[int, int]						,		__A							:   int      ):

				if is_complete(__A      ):
								return True

				for position in get_valid_pos(__A						,		len(__A      )      ):
								a_				,			a_		:  Dict											=      position

								if board[y][x] == 0:
												a_		:  Optional[Any]											=      curr + 1
												if open_knight_tour_helper(__A						,		__A						,		curr + 1      ):
																return True
												a_		:  Tuple											=      0

				return False





def 		_UpperCAmelCase   (							__A							:   int      ):
				a_		:  List[str]											=      [[0 for i in range(__A      )] for j in range(__A      )]

				for i in range(__A      ):
								for j in range(__A      ):
												a_		:  Optional[Any]											=      1
												if open_knight_tour_helper(__A						,		(i, j)						,		1      ):
																return board
												a_		:  Union[str, Any]											=      0

				a_		:  Dict											=      f'Open Kight Tour cannot be performed on a board of size {n}'
				raise ValueError(__A      )


if __name__ == "__main__":
		import doctest

		doctest.testmod()




