





import random

class SCREAMING_SNAKE_CASE_			:





					"""simple docstring"""


					def __init__(			self						,	_lowerCAmelCase						,	_lowerCAmelCase						):
										lowerCamelCase__      =							x
										lowerCamelCase__      =							y





					def        __magic_name__			(			self						):
										return (self.x**2 + self.y**2) <= 1






					@classmethod
					def        __magic_name__			(			cls						):
										return cls(x=random.random()						,	y=random.random()						)





def  __UpperCamelCase   (     a)			->float:
					if number_of_simulations < 1:
										raise ValueError("At least one simulation is necessary to estimate PI.")

					lowerCamelCase__      =							0
					for _ in range(a):
										lowerCamelCase__      =							Point.random_unit_square()

										if random_point.is_in_unit_circle():
															number_in_unit_circle += 1

					return 4 * number_in_unit_circle / number_of_simulations


if __name__ == "__main__":
						# import doctest

						# doctest.testmod()
						from math import pi

						A_        =  "Please enter the desired number of Monte Carlo simulations: "
						A_        =  estimate_pi(int(input(prompt).strip()))
						print(f"""An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}""")








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

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

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

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

					lowerCamelCase__      =							len(a)

					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(a)
										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(a) > 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]]))

