

from __future__ import annotations

from collections.abc import MutableSequence





class       snake_case_  :







			'''simple docstring'''

			def __init__(	self					:							Union[str, Any]				,  _UpperCamelCase					:							int				,  _UpperCamelCase					:							MutableSequence[float]				) ->None:
									if len(_UpperCamelCase				) != degree + 1:
															raise ValueError(
															    '''The number of coefficients should be equal to the degree + 1.'''				)

									snake_case_  =      list(_UpperCamelCase				)
									snake_case_  =      degree







			def __add__(	self					:							Any				,  _UpperCamelCase					:							Polynomial				) ->Polynomial:

									if self.degree > polynomial_a.degree:
															snake_case_  =      self.coefficients[:]
															for i in range(polynomial_a.degree + 1				):
																					coefficients[i] += polynomial_a.coefficients[i]
															return Polynomial(self.degree				,  _UpperCamelCase				)
									else:
															snake_case_  =      polynomial_a.coefficients[:]
															for i in range(self.degree + 1				):
																					coefficients[i] += self.coefficients[i]
															return Polynomial(polynomial_a.degree				,  _UpperCamelCase				)







			def __sub__(	self					:							Any				,  _UpperCamelCase					:							Polynomial				) ->Polynomial:
									return self + polynomial_a * Polynomial(0				,  [-1]				)







			def __neg__(	self					:							List[Any]				) ->Polynomial:
									return Polynomial(self.degree				,  [-c for c in self.coefficients]				)







			def __mul__(	self					:							List[Any]				,  _UpperCamelCase					:							Polynomial				) ->Polynomial:
									snake_case_  =      [0] * (self.degree + polynomial_a.degree + 1)
									for i in range(self.degree + 1				):
															for j in range(polynomial_a.degree + 1				):
																					coefficients[i + j] += (
																					    self.coefficients[i] * polynomial_a.coefficients[j]
																					)

									return Polynomial(self.degree + polynomial_a.degree				,  _UpperCamelCase				)







			def 							snake_case__(	self					:							Dict				,  _UpperCamelCase					:							int | float				) ->int | float:
									snake_case_  =      0
									for i in range(self.degree + 1				):
															result += self.coefficients[i] * (substitution**i)
									return result







			def __str__(	self					:							Tuple				) ->str:
									snake_case_  =      ''''''
									for i in range(self.degree				,  -1				,  -1				):
															if self.coefficients[i] == 0:
																					continue
															elif self.coefficients[i] > 0:
																					if polynomial:
																											polynomial += " + "
															else:
																					polynomial += " - "

															if i == 0:
																					polynomial += str(abs(self.coefficients[i]				)				)
															elif i == 1:
																					polynomial += str(abs(self.coefficients[i]				)				) + "x"
															else:
																					polynomial += str(abs(self.coefficients[i]				)				) + "x^" + str(_UpperCamelCase				)

									return polynomial







			def __repr__(	self					:							Union[str, Any]				) ->str:
									return self.__str__()







			def 							snake_case__(	self					:							Any				) ->Polynomial:
									snake_case_  =      [0] * self.degree
									for i in range(self.degree				):
															snake_case_  =      self.coefficients[i + 1] * (i + 1)
									return Polynomial(self.degree - 1				,  _UpperCamelCase				)







			def 							snake_case__(	self					:							List[Any]				,  _UpperCamelCase					:							int | float = 0				) ->Polynomial:
									snake_case_  =      [0] * (self.degree + 2)
									snake_case_  =      constant
									for i in range(self.degree + 1				):
															snake_case_  =      self.coefficients[i] / (i + 1)
									return Polynomial(self.degree + 1				,  _UpperCamelCase				)







			def __eq__(	self					:							str				,  _UpperCamelCase					:							object				) ->bool:
									if not isinstance(_UpperCamelCase				,  _UpperCamelCase				):
															return False

									if self.degree != polynomial_a.degree:
															return False

									for i in range(self.degree + 1				):
															if self.coefficients[i] != polynomial_a.coefficients[i]:
																					return False

									return True







			def __ne__(	self					:							str				,  _UpperCamelCase					:							object				) ->bool:
									return not self.__eq__(_UpperCamelCase				)

"""simple docstring"""


from __future__ import annotations






def    _snake_case					(		snake_case__ :  tuple[int, int]						,   snake_case__ :  int  ):
	A ,   A     = position
	A     = [
	    (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     = []

	for position in positions:
		A ,   A     = position
		if 0 <= y_test < n and 0 <= x_test < n:
			permissible_positions.append(snake_case__  )

	return permissible_positions






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






def    _snake_case					(		snake_case__ :  list[list[int]]						,   snake_case__ :  tuple[int, int]						,   snake_case__ :  int  ):

	if is_complete(snake_case__  ):
		return True

	for position in get_valid_pos(snake_case__						,   len(snake_case__  )  ):
		A ,   A     = position

		if board[y][x] == 0:
			A     = curr + 1
			if open_knight_tour_helper(snake_case__						,   snake_case__						,   curr + 1  ):
				return True
			A     = 0

	return False






def    _snake_case					(		snake_case__ :  int  ):
	A     = [[0 for i in range(snake_case__  )] for j in range(snake_case__  )]

	for i in range(snake_case__  ):
		for j in range(snake_case__  ):
			A     = 1
			if open_knight_tour_helper(snake_case__						,   (i, j)						,   1  ):
				return board
			A     = 0

	A     = F'Open Kight Tour cannot be performed on a board of size {n}'
	raise ValueError(snake_case__  )


if __name__ == "__main__":
		import doctest

		doctest.testmod()