'''simple docstring'''




def     snake_case__							(      _A: Tuple			)   ->     List[Any]:

		'''simple docstring'''


		if len(__UpperCamelCase			) == 0 or not all(len(__UpperCamelCase			) == 2 for pair in coordinates			):
				raise ValueError("""The program cannot work out a fitting polynomial."""			)

		if len({tuple(__UpperCamelCase			) for pair in coordinates}			) != len(__UpperCamelCase			):
				raise ValueError("""The program cannot work out a fitting polynomial."""			)

		lowerCAmelCase        =  {x for x, _ in coordinates}
		if len(__UpperCamelCase			) == 1:
				return f"x={coordinates[0][0]}"

		if len(__UpperCamelCase			) != len(__UpperCamelCase			):
				raise ValueError("""The program cannot work out a fitting polynomial."""			)

		lowerCAmelCase        =  len(__UpperCamelCase			)

		lowerCAmelCase        =  0
		lowerCAmelCase        =  []
		# put the x and x to the power values in a matrix
		while count_of_line < x:
				lowerCAmelCase        =  0
				lowerCAmelCase        =  coordinates[count_of_line][0]
				lowerCAmelCase        =  []
				while count_in_line < x:
						count_line.append(a ** (x - (count_in_line + 1))			)
						count_in_line += 1
				matrix.append(__UpperCamelCase			)
				count_of_line += 1

		lowerCAmelCase        =  0
		# put the y values into a vector
		lowerCAmelCase        =  []
		while count_of_line < x:
				vector.append(coordinates[count_of_line][1]			)
				count_of_line += 1

		lowerCAmelCase        =  0

		while count < x:
				lowerCAmelCase        =  0
				while zahlen < x:
						if count == zahlen:
								zahlen += 1
						if zahlen == x:
								break
						lowerCAmelCase        =  matrix[zahlen][count] / matrix[count][count]
						for counting_columns, item in enumerate(matrix[count]			):
								# manipulating all the values in the matrix
								matrix[zahlen][counting_columns] -= item * bruch
						# manipulating the values in the vector
						vector[zahlen] -= vector[count] * bruch
						zahlen += 1
				count += 1

		lowerCAmelCase        =  0
		# make solutions
		lowerCAmelCase        =  []
		while count < x:
				solution.append(str(vector[count] / matrix[count][count]			)			)
				count += 1

		lowerCAmelCase        =  0
		lowerCAmelCase        =  """f(x)="""

		while count < x:
				lowerCAmelCase        =  solution[count].split("""E"""			)
				if len(__UpperCamelCase			) > 1:
						lowerCAmelCase        =  f"{remove_e[0]}*10^{remove_e[1]}"
				solved += f"x^{x - (count + 1)}*{solution[count]}"
				if count + 1 != x:
						solved += "+"
				count += 1

		return solved


if __name__ == "__main__":
						print(points_to_polynomial([]))
						print(points_to_polynomial([[]]))
						print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
						print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
						print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
						print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
						print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
						print(points_to_polynomial([[1, 3], [2, 6], [3, 1_1]]))
						print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
						print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))





'''simple docstring'''




from __future__ import annotations

__lowercase				   =  [2, 1, 4, 5, 6, 0, 8, 9, 1, 2, 0, 6, 4, 2, 0, 6, 5, 3, 2, 7]






class     a__:


			'''simple docstring'''






			def __init__(       self	,   __lowerCAmelCase):




					"""simple docstring"""


					lowerCAmelCase        =  -1
					lowerCAmelCase        =  -1
					lowerCAmelCase        =  [-1] * length
					lowerCAmelCase        =  None
					lowerCAmelCase        =  None







			def __repr__(       self):




					"""simple docstring"""


					return f"Node(min_value={self.minn} max_value={self.maxx})"







def     snake_case__							(      _A: list[int]			)   ->     Node | None:

		'''simple docstring'''


		lowerCAmelCase        =  Node(len(_A			)			)
		lowerCAmelCase ,					lowerCAmelCase        =  min(_A			), max(_A			)
		# Leaf node case where the node contains only one unique value
		if root.minn == root.maxx:
				return root
		lowerCAmelCase        =  (root.minn + root.maxx) // 2

		lowerCAmelCase        =  []
		lowerCAmelCase        =  []

		for index, num in enumerate(_A			):
				if num <= pivot:
						left_arr.append(_A			)
				else:
						right_arr.append(_A			)
				lowerCAmelCase        =  len(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		lowerCAmelCase        =  build_tree(_A			)
		return root






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if index < 0 or node is None:
				return 0
		# Leaf node cases
		if node.minn == node.maxx:
				return index + 1 if node.minn == num else 0
		lowerCAmelCase        =  (node.minn + node.maxx) // 2
		if num <= pivot:
				# go the left subtree and map index to the left subtree
				return rank_till_index(node.left				,       _A				,       node.map_left[index] - 1			)
		else:
				# go to the right subtree and map index to the right subtree
				return rank_till_index(node.right				,       _A				,       index - node.map_left[index]			)






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if start > end:
				return 0
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       _A			)
		lowerCAmelCase        =  rank_till_index(_A				,       _A				,       start - 1			)
		return rank_till_end - rank_before_start






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if index > (end - start) or start > end or node is None:
				return -1
		# Leaf node case
		if node.minn == node.maxx:
				return node.minn
		# Number of elements in the left subtree in interval [start, end]
		lowerCAmelCase        =  node.map_left[end] - (
		    node.map_left[start - 1] if start else 0
		)
		if num_elements_in_left_tree > index:
				return quantile(
				    node.left				,       _A				,       (node.map_left[start - 1] if start else 0)				,       node.map_left[end] - 1				,       )
		else:
				return quantile(
				    node.right				,       index - num_elements_in_left_tree				,       start - (node.map_left[start - 1] if start else 0)				,       end - node.map_left[end]				,       )






def     snake_case__							(      _A: Node | None				,       _A: int				,       _A: int				,       _A: int				,       _A: int			)   ->     int:

		'''simple docstring'''


		if (
		    start > end
		    or node is None
		    or start_num > end_num
		    or node.minn > end_num
		    or node.maxx < start_num
		):
				return 0
		if start_num <= node.minn and node.maxx <= end_num:
				return end - start + 1
		lowerCAmelCase        =  range_counting(
		    node.left				,       (node.map_left[start - 1] if start else 0)				,       node.map_left[end] - 1				,       _A				,       _A				,       )
		lowerCAmelCase        =  range_counting(
		    node.right				,       start - (node.map_left[start - 1] if start else 0)				,       end - node.map_left[end]				,       _A				,       _A				,       )
		return left + right


if __name__ == "__main__":
						import doctest

						doctest.testmod()




