




from __future__ import annotations

from collections.abc import Generator
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    if not isinstance(UpperCAmelCase_ ,     UpperCAmelCase_		) or n < 1:
        raise Exception("""Sequence only defined for positive integers"""		)

    yield n
    while n != 1:
        if n % 2 == 0:
            n //= 2
        else:
            a__  			=							3 * n + 1
        yield n
def 							__lowerCAmelCase			():
    a__  			=							43
    a__  			=							tuple(collatz_sequence(UpperCAmelCase_		)		)
    print(UpperCAmelCase_		)
    print(F"""Collatz sequence from {n} took {len(UpperCAmelCase_		)} steps."""		)


if __name__ == "__main__":
 main()












from itertools import count
from math import asin, pi, sqrt
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    return (
        (1 - 2 * point) * sqrt(point - point**2		) + 2 * point + asin(sqrt(1 - point		)		)
    ) / 4
def 							__lowerCAmelCase			(UpperCAmelCase_		):
    a__  			=							(circles_number + 1 - sqrt(2 * circles_number		)) / (
        2 * (circles_number**2 + 1)
    )
    a__  			=							circles_number * intersection_y

    a__  			=							intersection_x * intersection_y / 2
    a__  			=							circle_bottom_arc_integral(
        1 / 2		) - circle_bottom_arc_integral(UpperCAmelCase_		)

    return triangle_area + concave_region_area
def 							__lowerCAmelCase			(UpperCAmelCase_ = 1 / 1_000		):
    a__  			=							(1 - pi / 4) / 4

    for n in count(1		):
        if concave_triangle_area(UpperCAmelCase_		) / l_section_area < fraction:
            return n

    return -1


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






