


'''simple docstring'''



from __future__ import annotations

from collections.abc import Callable

__A		=list[list[float | int]]







def       _UpperCamelCase (						UpperCamelCase__	, UpperCamelCase__   ):
			UpperCAmelCase__					:   int       				=       len(UpperCamelCase__   )
			UpperCAmelCase__					:   Matrix       				=       [[0 for _ in range(size + 1   )] for _ in range(UpperCamelCase__   )]
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   float

			for row in range(UpperCamelCase__   ):
						for col in range(UpperCamelCase__   ):
									UpperCAmelCase__					:   Optional[int]       				=       matrix[row][col]

						UpperCAmelCase__					:   Dict       				=       vector[row][0]

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

						for rowa in range(row + 1	, UpperCamelCase__   ):
									UpperCAmelCase__					:   List[str]       				=       augmented[rowa][col] / augmented[row][col]
									UpperCAmelCase__					:   Dict       				=       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__   ):
									UpperCAmelCase__					:   List[str]       				=       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]	, 1_0   )] for row in range(UpperCamelCase__   )
			]







def       _UpperCamelCase (						UpperCamelCase__   ):
			UpperCAmelCase__					:   int       				=       len(UpperCamelCase__   )
			UpperCAmelCase__					:   Matrix       				=       [[0 for _ in range(UpperCamelCase__   )] for _ in range(UpperCamelCase__   )]
			UpperCAmelCase__					:   Matrix       				=       [[0] for _ in range(UpperCamelCase__   )]
			UpperCAmelCase__					:   Matrix
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   int
			UpperCAmelCase__					:   int

			for x_val, y_val in enumerate(UpperCamelCase__   ):
						for col in range(UpperCamelCase__   ):
									UpperCAmelCase__					:   Dict       				=       (x_val + 1) ** (size - col - 1)
						UpperCAmelCase__					:   Any       				=       y_val

			UpperCAmelCase__					:   List[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       _UpperCamelCase (						UpperCamelCase__   ):
			return (
			    1
			    - variable
			    + variable**2
			    - variable**3
			    + variable**4
			    - variable**5
			    + variable**6
			    - variable**7
			    + variable**8
			    - variable**9
			    + variable**1_0
			)







def       _UpperCamelCase (						UpperCamelCase__ = question_function	, UpperCamelCase__ = 1_0   ):
			UpperCAmelCase__					:   list[int]       				=       [func(UpperCamelCase__   ) for x_val in range(1	, order + 1   )]

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

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

			for poly in polynomials:
						UpperCAmelCase__					:   Tuple       				=       1
						while func(UpperCamelCase__   ) == poly(UpperCamelCase__   ):
									x_val += 1

						ret += poly(UpperCamelCase__   )

			return ret


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



'''simple docstring'''



import colorsys

from PIL import Image  # type: ignore







def       _UpperCamelCase (						UpperCamelCase__	, UpperCamelCase__	, UpperCamelCase__   ):
			UpperCAmelCase__					:   Optional[Any]       				=       x
			UpperCAmelCase__					:   Optional[int]       				=       y
			for step in range(UpperCamelCase__   ):  # noqa: B007
						UpperCAmelCase__					:   List[str]       				=       a * a - b * b + x
						UpperCAmelCase__					:   Optional[int]       				=       2 * a * b + y
						UpperCAmelCase__					:   int       				=       a_new

						# divergence happens for all complex number with an absolute value
						# greater than 4
						if a * a + b * b > 4:
									break
			return step / (max_step - 1)







def       _UpperCamelCase (						UpperCamelCase__   ):
			if distance == 1:
						return (0, 0, 0)
			else:
						return (2_5_5, 2_5_5, 2_5_5)







def       _UpperCamelCase (						UpperCamelCase__   ):
			if distance == 1:
						return (0, 0, 0)
			else:
						return tuple(round(i * 2_5_5   ) for i in colorsys.hsv_to_rgb(UpperCamelCase__	, 1	, 1   )   )







def       _UpperCamelCase (						UpperCamelCase__ = 8_0_0	, UpperCamelCase__ = 6_0_0	, UpperCamelCase__ = -0.6	, UpperCamelCase__ = 0	, UpperCamelCase__ = 3.2	, UpperCamelCase__ = 5_0	, UpperCamelCase__ = True	, ):
			UpperCAmelCase__					:   str       				=       Image.new("""RGB"""	, (image_width, image_height)   )
			UpperCAmelCase__					:   Optional[int]       				=       img.load()

			# loop through the image-coordinates
			for image_x in range(UpperCamelCase__   ):
						for image_y in range(UpperCamelCase__   ):
									# determine the figure-coordinates based on the image-coordinates
									UpperCAmelCase__					:   Union[str, Any]       				=       figure_width / image_width * image_height
									UpperCAmelCase__					:   Union[str, Any]       				=       figure_center_x + (image_x / image_width - 0.5) * figure_width
									UpperCAmelCase__					:   Tuple       				=       figure_center_y + (image_y / image_height - 0.5) * figure_height

									UpperCAmelCase__					:   List[str]       				=       get_distance(UpperCamelCase__	, UpperCamelCase__	, UpperCamelCase__   )

									# color the corresponding pixel based on the selected coloring-function
									if use_distance_color_coding:
												UpperCAmelCase__					:   Any       				=       get_color_coded_rgb(UpperCamelCase__   )
									else:
												UpperCAmelCase__					:   Tuple       				=       get_black_and_white_rgb(UpperCamelCase__   )

			return img


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		# colored version, full figure
		__A		=get_image()

		# uncomment for colored version, different section, zoomed in
		# img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
		# figure_width = 0.8)

		# uncomment for black and white version, full figure
		# img = get_image(use_distance_color_coding = False)

		# uncomment to save the image
		# img.save("mandelbrot.png")

		img.show()