"""Binary search helpers returning insertion positions.

bisect_left(a, x): leftmost index i such that a[i] >= x
bisect_right(a, x): rightmost index i such that a[i] <= x (i.e. insertion point after equals)
Both assume a sorted ascending.
"""
from __future__ import annotations


def bisect_left(a, x):
    lo, hi = 0, len(a)
    while lo < hi:
        mid = (lo + hi) // 2
        if a[mid] < x:
            lo = mid + 1
        else:
            hi = mid
    return lo


def bisect_right(a, x):
    lo, hi = 0, len(a)
    while lo < hi:
        mid = (lo + hi) // 2
        if a[mid] <= x:
            lo = mid + 1
        else:
            hi = mid
    return lo


def contains(a, x):
    i = bisect_left(a, x)
    return i < len(a) and a[i] == x