import typing


def yep(s: str) -> bool:
    return s.strip().lower().startswith("y")


def guess_bisect_repl(lower: int, upper: int) -> int:
    mid = lower + ((upper - lower) // 2)

    if yep(input(f"is it {mid}? ")):
        return mid

    if yep(input(f"higher than {mid}? ")):
        return guess_bisect_repl(mid, upper)

    return guess_bisect_repl(lower, mid)


def find_sqrt_ish(n: int) -> int:
    return int(find_bisect(0, n, gen_sqrt_check(n)))


def gen_sqrt_check(n: int) -> typing.Callable[[float], int]:
    def check(mid: float) -> int:
        mid_sq: float = mid * mid

        if mid_sq == n:
            return 0

        if mid_sq < n:
            return 1

        return -1

    return check


def find_bisect(lower: float, upper: float, check: typing.Callable[[float], int]) -> float:
    mid: float = lower + ((upper - lower) / 2)

    print(f"lower={lower} mid={mid} upper={upper}")

    if mid == lower or mid == upper or check(mid) == 0:
        return mid

    if check(mid) == 1:
        return find_bisect(mid, upper, check)

    return find_bisect(lower, mid, check)


def cartesian_path(p0: tuple[int, int], p1: tuple[int, int]) -> list[tuple[int, int]]:
    path: list[tuple[int, int]] = []

    if p0 < p1:
        for i in range(p0[1], p1[1]):
            path.append((i, p0[0]))

        for i in range(p0[0], p1[0]):
            path.append((p1[1], i))

    else:
        for i in range(p0[1], p1[1] - 1, -1):
            path.append((i, p0[0]))

        for i in range(p0[0] - 1, p1[0], -1):
            path.append((p1[1], i))

    return path


def gen_matrix(width: int, height: int) -> list[list[int]]:
    return [list(range(width)) for _ in range(height)]