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)]


class Roman:
    SIMPLE = {
        "I": 1,
        "V": 5,
        "X": 10,
        "L": 50,
        "C": 100,
        "D": 500,
        "M": 1000,
    }
    SIMPLE_REVERSE = {v: k for k, v in SIMPLE.items()}
    COMPOUND = {
        "IV": 4,
        "IX": 9,
        "XL": 40,
        "XC": 90,
        "CD": 400,
        "CM": 900,
    }
    PREFIXES = {k[0] for k in COMPOUND.keys()}
    COMPOUND_REVERSE = {v: k for k, v in COMPOUND.items()}
    ALL = SIMPLE | COMPOUND
    ALL_REVERSE = {v: k for k, v in ALL.items()}

    @classmethod
    def i2r(cls, i: int) -> str:
        if i > 100_000:
            raise ValueError(f"{i} is too silly")

        r: list[str] = []

        for int_val, roman_val in sorted(cls.ALL_REVERSE.items(), reverse=True):
            remainder = i % int_val

            r += [roman_val] * int((i - remainder) / int_val)

            i = remainder

        return "".join(r)

    @classmethod
    def r2i(cls, r: str) -> int:
        total = 0
        offset = 0

        for i in range(len(r)):
            if i + offset > len(r) - 1:
                break

            c = r[i + offset]
            if (
                c in cls.PREFIXES
                and (i + offset + 1) < len(r)
                and c + r[i + offset + 1] in cls.ALL
            ):
                total += cls.ALL[c + r[i + offset + 1]]
                offset += 1
                continue

            total += cls.ALL[c]

        return total