"""simple docstring"""







from __future__ import annotations

from collections.abc import Callable

lowerCamelCase__           =	list[list[float | int]]
def 					__lowerCAmelCase       (_UpperCamelCase ,						_UpperCamelCase	):
	__lowerCAmelCase    :		int		    =   len(_UpperCamelCase	)
	__lowerCAmelCase    :		Matrix		    =   [[0 for _ in range(size + 1	)] for _ in range(_UpperCamelCase	)]
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		float

	for row in range(_UpperCamelCase	):
		for col in range(_UpperCamelCase	):
			__lowerCAmelCase    :		Union[str, Any]		    =   matrix[row][col]

		__lowerCAmelCase    :		Optional[int]		    =   vector[row][0]

	__lowerCAmelCase    :		Union[str, Any]		    =   0
	__lowerCAmelCase    :		str		    =   0
	while row < size and col < size:
		# pivoting
		__lowerCAmelCase    :		Tuple		    =   max((abs(augmented[rowa][col]	), rowa) for rowa in range(_UpperCamelCase ,						_UpperCamelCase	)	)[
		    1
		]
		if augmented[pivot_row][col] == 0:
			col += 1
			continue
		else:
			__lowerCAmelCase    :		Optional[Any]		    =   augmented[pivot_row], augmented[row]

		for rowa in range(row + 1 ,						_UpperCamelCase	):
			__lowerCAmelCase    :		Tuple		    =   augmented[rowa][col] / augmented[row][col]
			__lowerCAmelCase    :		Optional[Any]		    =   0
			for cola in range(col + 1 ,						size + 1	):
				augmented[rowa][cola] -= augmented[row][cola] * ratio

		row += 1
		col += 1

	# back substitution
	for col in range(1 ,						_UpperCamelCase	):
		for row in range(_UpperCamelCase	):
			__lowerCAmelCase    :		Optional[Any]		    =   augmented[row][col] / augmented[col][col]
			for cola in range(_UpperCamelCase ,						size + 1	):
				augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
	return [
	    [round(augmented[row][size] / augmented[row][row] ,						10	)] for row in range(_UpperCamelCase	)
	]
def 					__lowerCAmelCase       (_UpperCamelCase	):
	__lowerCAmelCase    :		int		    =   len(_UpperCamelCase	)
	__lowerCAmelCase    :		Matrix		    =   [[0 for _ in range(_UpperCamelCase	)] for _ in range(_UpperCamelCase	)]
	__lowerCAmelCase    :		Matrix		    =   [[0] for _ in range(_UpperCamelCase	)]
	__lowerCAmelCase    :		Matrix
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		int
	__lowerCAmelCase    :		int

	for x_val, y_val in enumerate(_UpperCamelCase	):
		for col in range(_UpperCamelCase	):
			__lowerCAmelCase    :		List[Any]		    =   (x_val + 1) ** (size - col - 1)
		__lowerCAmelCase    :		int		    =   y_val

	__lowerCAmelCase    :		Union[str, Any]		    =   solve(_UpperCamelCase ,						_UpperCamelCase	)

	def interpolated_func(_UpperCamelCase	) -> int:
		return sum(
		    round(coeffs[x_val][0]	) * (var ** (size - x_val - 1))
		    for x_val in range(_UpperCamelCase	)	)

	return interpolated_func
def 					__lowerCAmelCase       (_UpperCamelCase	):
	return (
	    1
	    - variable
	    + variable**2
	    - variable**3
	    + variable**4
	    - variable**5
	    + variable**6
	    - variable**7
	    + variable**8
	    - variable**9
	    + variable**10
	)
def 					__lowerCAmelCase       (_UpperCamelCase = question_function ,						_UpperCamelCase = 10	):
	__lowerCAmelCase    :		list[int]		    =   [func(_UpperCamelCase	) for x_val in range(1 ,						order + 1	)]

	__lowerCAmelCase    :		list[Callable[[int], int]]		    =   [
	    interpolate(data_points[:max_coeff]	) for max_coeff in range(1 ,						order + 1	)
	]

	__lowerCAmelCase    :		int		    =   0
	__lowerCAmelCase    :		Callable[[int], int]
	__lowerCAmelCase    :		int

	for poly in polynomials:
		__lowerCAmelCase    :		Optional[Any]		    =   1
		while func(_UpperCamelCase	) == poly(_UpperCamelCase	):
			x_val += 1

		ret += poly(_UpperCamelCase	)

	return ret


if __name__ == "__main__":
							print(f'{solution() = }')

"""simple docstring"""







class 				A__ :

		def __init__(   self				):
			__lowerCAmelCase    :		Optional[int]		    =   0
			__lowerCAmelCase    :		Any		    =   0
			__lowerCAmelCase    :		List[Any]		    =   {}

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE				):
			if vertex not in self.adjacency:
				__lowerCAmelCase    :		Dict		    =   {}
				self.num_vertices += 1

		def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
			self.add_vertex(_SCREAMING_SNAKE_CASE				)
			self.add_vertex(_SCREAMING_SNAKE_CASE				)

			if head == tail:
				return

			__lowerCAmelCase    :		Union[str, Any]		    =   weight
			__lowerCAmelCase    :		str		    =   weight

		def 							__lowerCamelCase							(   self				):
			__lowerCAmelCase    :		int		    =   self.get_edges()
			for edge in edges:
				__lowerCAmelCase    ,     __lowerCAmelCase    ,     __lowerCAmelCase    :		Optional[Any]		    =   edge
				edges.remove((tail, head, weight)				)
			for i in range(len(_SCREAMING_SNAKE_CASE				)				):
				__lowerCAmelCase    :		List[str]		    =   list(edges[i]				)

			edges.sort(key=lambda _SCREAMING_SNAKE_CASE				: e[2]				)
			for i in range(len(_SCREAMING_SNAKE_CASE				) - 1				):
				if edges[i][2] >= edges[i + 1][2]:
					__lowerCAmelCase    :		Dict		    =   edges[i][2] + 1
			for edge in edges:
				__lowerCAmelCase    ,     __lowerCAmelCase    ,     __lowerCAmelCase    :		str		    =   edge
				__lowerCAmelCase    :		Union[str, Any]		    =   weight
				__lowerCAmelCase    :		Optional[int]		    =   weight

		def __str__(   self				):
			__lowerCAmelCase    :		List[Any]		    =   ''
			for tail in self.adjacency:
				for head in self.adjacency[tail]:
					__lowerCAmelCase    :		str		    =   self.adjacency[head][tail]
					string += f"{head} -> {tail} == {weight}\n"
			return string.rstrip('\n'				)

		def 							__lowerCamelCase							(   self				):
			__lowerCAmelCase    :		Tuple		    =   []
			for tail in self.adjacency:
				for head in self.adjacency[tail]:
					output.append((tail, head, self.adjacency[head][tail])				)
			return output

		def 							__lowerCamelCase							(   self				):
			return self.adjacency.keys()

		@staticmethod
		def 							__lowerCamelCase							(   _SCREAMING_SNAKE_CASE=None ,	_SCREAMING_SNAKE_CASE=None				):
			__lowerCAmelCase    :		List[str]		    =   Graph()
			if vertices is None:
				__lowerCAmelCase    :		int		    =   []
			if edges is None:
				__lowerCAmelCase    :		Optional[int]		    =   []
			for vertex in vertices:
				g.add_vertex(_SCREAMING_SNAKE_CASE				)
			for edge in edges:
				g.add_edge(*_SCREAMING_SNAKE_CASE				)
			return g




		class 				A__ :

				def __init__(   self				):
					__lowerCAmelCase    :		Dict		    =   {}
					__lowerCAmelCase    :		Optional[int]		    =   {}

				def __len__(   self				):
					return len(self.parent				)

				def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE				):
					if item in self.parent:
						return self.find(_SCREAMING_SNAKE_CASE				)

					__lowerCAmelCase    :		Dict		    =   item
					__lowerCAmelCase    :		int		    =   0
					return item

				def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE				):
					if item not in self.parent:
						return self.make_set(_SCREAMING_SNAKE_CASE				)
					if item != self.parent[item]:
						__lowerCAmelCase    :		Dict		    =   self.find(self.parent[item]				)
					return self.parent[item]







				def 							__lowerCamelCase							(   self ,	_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				):
					__lowerCAmelCase    :		Any		    =   self.find(_SCREAMING_SNAKE_CASE				)
					__lowerCAmelCase    :		List[str]		    =   self.find(_SCREAMING_SNAKE_CASE				)

					if roota == roota:
						return roota

					if self.rank[roota] > self.rank[roota]:
						__lowerCAmelCase    :		Tuple		    =   roota
						return roota

					if self.rank[roota] < self.rank[roota]:
						__lowerCAmelCase    :		str		    =   roota
						return roota

					if self.rank[roota] == self.rank[roota]:
						self.rank[roota] += 1
						__lowerCAmelCase    :		Optional[int]		    =   roota
						return roota
					return None



		@staticmethod
		def 							__lowerCamelCase							(   _SCREAMING_SNAKE_CASE				):
			__lowerCAmelCase    :		Union[str, Any]		    =   graph.num_vertices

			__lowerCAmelCase    :		List[Any]		    =   Graph.UnionFind()
			__lowerCAmelCase    :		Tuple		    =   []
			while num_components > 1:
				__lowerCAmelCase    :		str		    =   {}
				for vertex in graph.get_vertices():
					__lowerCAmelCase    :		Union[str, Any]		    =   -1

				__lowerCAmelCase    :		int		    =   graph.get_edges()
				for edge in edges:
					__lowerCAmelCase    ,     __lowerCAmelCase    ,     __lowerCAmelCase    :		Union[str, Any]		    =   edge
					edges.remove((tail, head, weight)				)
				for edge in edges:
					__lowerCAmelCase    ,     __lowerCAmelCase    ,     __lowerCAmelCase    :		Any		    =   edge
					__lowerCAmelCase    :		Optional[int]		    =   union_find.find(_SCREAMING_SNAKE_CASE				)
					__lowerCAmelCase    :		str		    =   union_find.find(_SCREAMING_SNAKE_CASE				)
					if seta != seta:
						if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
							__lowerCAmelCase    :		List[str]		    =   [head, tail, weight]

						if cheap_edge[seta] == -1 or cheap_edge[seta][2] > weight:
							__lowerCAmelCase    :		List[str]		    =   [head, tail, weight]
				for vertex in cheap_edge:
					if cheap_edge[vertex] != -1:
						__lowerCAmelCase    ,     __lowerCAmelCase    ,     __lowerCAmelCase    :		str		    =   cheap_edge[vertex]
						if union_find.find(_SCREAMING_SNAKE_CASE				) != union_find.find(_SCREAMING_SNAKE_CASE				):
							union_find.union(_SCREAMING_SNAKE_CASE ,	_SCREAMING_SNAKE_CASE				)
							mst_edges.append(cheap_edge[vertex]				)
							__lowerCAmelCase    :		List[str]		    =   num_components - 1
			__lowerCAmelCase    :		Union[str, Any]		    =   Graph.build(edges=_SCREAMING_SNAKE_CASE				)
			return mst