struct convex_hull {
    int n, m;
    vector<int> on_hull;
    // WARN: counter-clockwise
    vector<int> hull;

    // WARN: sort the points before initializing
    template <typename T>
    convex_hull(const vector<point<T>>& a) : n(a.size()), on_hull(n) {
        for (int i = 0; i < n; ++i) {
            int m = hull.size();
            // WARN: keeping the points that are on edges
            while (m > 1 and (a[hull[m - 1]] - a[hull[m - 2]]) * (a[i] - a[hull[m - 2]]) < 0) {
                on_hull[hull[m - 1]] -= 1;
                hull.pop_back();
                m -= 1;
            }
            on_hull[i] += 1;
            hull.emplace_back(i);
        }
        int low = hull.size();

        for (int i = n - 2; ~i; --i) {
            int m = hull.size();
            while (m > low and (a[hull[m - 1]] - a[hull[m - 2]]) * (a[i] - a[hull[m - 2]]) < 0) {
                on_hull[hull[m - 1]] -= 1;
                hull.pop_back();
                m -= 1;
            }
            on_hull[i] += 1;
            hull.emplace_back(i);
        }
        if (n > 1) {
            hull.pop_back();
        }
        m = hull.size();
    }
};