

from __future__ import annotations

from functools import lru_cache
from math import ceil

lowercase_			=  100

lowercase_			=  set(range(3, NUM_PRIMES, 2))
primes.add(2)
lowercase_			=  42

for prime in range(3, ceil(NUM_PRIMES**0.5), 2):
     if prime not in primes:
          continue
     primes.difference_update(set(range(prime * prime, NUM_PRIMES, prime)))
@lru_cache(maxsize=100				)
def 						_snake_case(   SCREAMING_SNAKE_CASE__      :     int				)							->				set[int]:





    '''simple docstring'''






    if number_to_partition < 0:
        return set()
    elif number_to_partition == 0:
        return {1}

    A__   				=       set()
    A__   				=       42
    A__   				=       42

    for prime in primes:
        if prime > number_to_partition:
            continue
        for sub in partition(number_to_partition - prime				):
            ret.add(sub * prime				)

    return ret
def 						_snake_case(   SCREAMING_SNAKE_CASE__      :     int = 5000				)							->				int | None:





    '''simple docstring'''






    for number_to_partition in range(1   ,      SCREAMING_SNAKE_CASE__				):
        if len(partition(SCREAMING_SNAKE_CASE__				)				) > number_unique_partitions:
            return number_to_partition
    return None


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




from __future__ import annotations

from scipy.special import comb  # type: ignore


class 			A						:







    """simple docstring"""





    def __init__(				self			:	Any,lowercase_			:	list[tuple[float, float]]		)->   Optional[int]:





        '''simple docstring'''







        A__   				=       list_of_points
        # Degree determines the flexibility of the curve.
        # Degree = 1 will produce a straight line.
        A__   				=       len(lowercase_		) - 1


    def 						snake_case__					(				self			:	List[Any],lowercase_			:	float		)->   list[float]:





        '''simple docstring'''







        assert 0 <= t <= 1, "Time t must be between 0 and 1."
        A__   				=       []
        for i in range(len(self.list_of_points		)		):
            # basis function for each i
            output_values.append(
                comb(self.degree,lowercase_		) * ((1 - t) ** (self.degree - i)) * (t**i)		)
        # the basis must sum up to 1 for it to produce a valid Bezier curve.
        assert round(sum(lowercase_		),5		) == 1
        return output_values


    def 						snake_case__					(				self			:	str,lowercase_			:	float		)->   tuple[float, float]:





        '''simple docstring'''







        assert 0 <= t <= 1, "Time t must be between 0 and 1."

        A__   				=       self.basis_function(lowercase_		)
        A__   				=       0.0
        A__   				=       0.0
        for i in range(len(self.list_of_points		)		):
            # For all points, sum up the product of i-th basis function and i-th point.
            x += basis_function[i] * self.list_of_points[i][0]
            y += basis_function[i] * self.list_of_points[i][1]
        return (x, y)


    def 						snake_case__					(				self			:	str,lowercase_			:	float = 0.01		)->   str:





        '''simple docstring'''







        from matplotlib import pyplot as plt  # type: ignore

        A__   				=       []  # x coordinates of points to plot
        A__   				=       []  # y coordinates of points to plot

        A__   				=       0.0
        while t <= 1:
            A__   				=       self.bezier_curve_function(lowercase_		)
            to_plot_x.append(value[0]		)
            to_plot_y.append(value[1]		)
            t += step_size

        A__   				=       [i[0] for i in self.list_of_points]
        A__   				=       [i[1] for i in self.list_of_points]

        plt.plot(
            lowercase_,lowercase_,color='blue',label='Curve of Degree ' + str(self.degree		),)
        plt.scatter(lowercase_,lowercase_,color='red',label='Control Points'		)
        plt.legend()
        plt.show()


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     BezierCurve([(1, 2), (3, 5)]).plot_curve()  # degree 1
     BezierCurve([(0, 0), (5, 5), (5, 0)]).plot_curve()  # degree 2
     BezierCurve([(0, 0), (5, 5), (5, 0), (2.5, -2.5)]).plot_curve()  # degree 3

