// MaxFlow based and AtCoder Library, single class, no namespace, no private variables, compatible
// with C++11 Reference: <https://atcoder.github.io/ac-library/production/document_ja/maxflow.html>
template <class Cap> struct mf_graph {
    struct simple_queue_int {
        std::vector<int> payload;
        int pos = 0;
        void reserve(int n) { payload.reserve(n); }
        int size() const { return int(payload.size()) - pos; }
        bool empty() const { return pos == int(payload.size()); }
        void push(const int &t) { payload.push_back(t); }
        int &front() { return payload[pos]; }
        void clear() {
            payload.clear();
            pos = 0;
        }
        void pop() { pos++; }
    };
 
    mf_graph() : _n(0) {}
    mf_graph(int n) : _n(n), g(n) {}
 
    int add_edge(int from, int to, Cap cap) {
        assert(0 <= from && from < _n);
        assert(0 <= to && to < _n);
        assert(0 <= cap);
        int m = int(pos.size());
        pos.push_back({from, int(g[from].size())});
        int from_id = int(g[from].size());
        int to_id = int(g[to].size());
        if (from == to) to_id++;
        g[from].push_back(_edge{to, to_id, cap});
        g[to].push_back(_edge{from, from_id, 0});
        return m;
    }
 
    struct edge {
        int from, to;
        Cap cap, flow;
    };
 
    edge get_edge(int i) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        auto _e = g[pos[i].first][pos[i].second];
        auto _re = g[_e.to][_e.rev];
        return edge{pos[i].first, _e.to, _e.cap + _re.cap, _re.cap};
    }
    std::vector<edge> edges() {
        int m = int(pos.size());
        std::vector<edge> result;
        for (int i = 0; i < m; i++) { result.push_back(get_edge(i)); }
        return result;
    }
    void change_edge(int i, Cap new_cap, Cap new_flow) {
        int m = int(pos.size());
        assert(0 <= i && i < m);
        assert(0 <= new_flow && new_flow <= new_cap);
        auto &_e = g[pos[i].first][pos[i].second];
        auto &_re = g[_e.to][_e.rev];
        _e.cap = new_cap - new_flow;
        _re.cap = new_flow;
    }
 
    std::vector<int> level, iter;
    simple_queue_int que;
 
    void _bfs(int s, int t) {
        std::fill(level.begin(), level.end(), -1);
        level[s] = 0;
        que.clear();
        que.push(s);
        while (!que.empty()) {
            int v = que.front();
            que.pop();
            for (auto e : g[v]) {
                if (e.cap == 0 || level[e.to] >= 0) continue;
                level[e.to] = level[v] + 1;
                if (e.to == t) return;
                que.push(e.to);
            }
        }
    }
    Cap _dfs(int v, int s, Cap up) {
        if (v == s) return up;
        Cap res = 0;
        int level_v = level[v];
        for (int &i = iter[v]; i < int(g[v].size()); i++) {
            _edge &e = g[v][i];
            if (level_v <= level[e.to] || g[e.to][e.rev].cap == 0) continue;
            Cap d = _dfs(e.to, s, std::min(up - res, g[e.to][e.rev].cap));
            if (d <= 0) continue;
            g[v][i].cap += d;
            g[e.to][e.rev].cap -= d;
            res += d;
            if (res == up) return res;
        }
        level[v] = _n;
        return res;
    }
 
    Cap flow(int s, int t) { return flow(s, t, std::numeric_limits<Cap>::max()); }
    Cap flow(int s, int t, Cap flow_limit) {
        assert(0 <= s && s < _n);
        assert(0 <= t && t < _n);
        assert(s != t);
 
        level.assign(_n, 0), iter.assign(_n, 0);
        que.clear();
 
        Cap flow = 0;
        while (flow < flow_limit) {
            _bfs(s, t);
            if (level[t] == -1) break;
            std::fill(iter.begin(), iter.end(), 0);
            Cap f = _dfs(t, s, flow_limit - flow);
            if (!f) break;
            flow += f;
        }
        return flow;
    }
 
    std::vector<bool> min_cut(int s) {
        std::vector<bool> visited(_n);
        simple_queue_int que;
        que.push(s);
        while (!que.empty()) {
            int p = que.front();
            que.pop();
            visited[p] = true;
            for (auto e : g[p]) {
                if (e.cap && !visited[e.to]) {
                    visited[e.to] = true;
                    que.push(e.to);
                }
            }
        }
        return visited;
    }
 
    void dump_graphviz(std::string filename = "maxflow") const {
        std::ofstream ss(filename + ".DOT");
        ss << "digraph{\n";
        for (int i = 0; i < _n; i++) {
            for (const auto &e : g[i]) {
                if (e.cap > 0) ss << i << "->" << e.to << "[label=" << e.cap << "];\n";
            }
        }
        ss << "}\n";
        ss.close();
        return;
    }
 
    int _n;
    struct _edge {
        int to, rev;
        Cap cap;
    };
    std::vector<std::pair<int, int>> pos;
    std::vector<std::vector<_edge>> g;
};
 
 
// MaxFlow with lower bound
// https://snuke.hatenablog.com/entry/2016/07/10/043918
// https://ei1333.github.io/library/graph/flow/maxflow-lower-bound.cpp
// flush(s, t): Calculate maxflow (if solution exists), -1 (otherwise)
template <typename Cap> struct MaxFlowLowerBound {
    using Maxflow = mf_graph<Cap>;
    int N;
    Maxflow mf;
    std::vector<Cap> in;
    MaxFlowLowerBound(int N = 0) : N(N), mf(N + 2), in(N) {}
 
    std::vector<Cap> cap_lo_info;
 
    int add_edge(int from, int to, Cap cap_lo, Cap cap_hi) {
        assert(0 <= from and from < N);
        assert(0 <= to and to < N);
        assert(0 <= cap_lo and cap_lo <= cap_hi);
        in[from] -= cap_lo;
        in[to] += cap_lo;
        cap_lo_info.push_back(cap_lo);
        return mf.add_edge(from, to, cap_hi - cap_lo);
    }
 
    Cap flow(int s, int t) {
        assert(s != t);
        assert(0 <= s and s < N);
        assert(0 <= t and t < N);
        Cap sum = 0;
        for (int i = 0; i < N; i++) {
            if (in[i] > 0) mf.add_edge(N, i, in[i]), sum += in[i];
            if (in[i] < 0) mf.add_edge(i, N + 1, -in[i]);
        }
        mf.add_edge(t, s, std::numeric_limits<Cap>::max());
        if (mf.flow(N, N + 1) < sum) return -1;
        return mf.flow(s, t);
    }
 
    typename Maxflow::edge get_edge(int i) {
        auto ret = mf.get_edge(i);
        ret.cap += cap_lo_info.at(i);
        ret.flow += cap_lo_info.at(i);
        return ret;
    }
 
    std::vector<typename Maxflow::edge> edges() {
        std::vector<typename Maxflow::edge> result;
        for (int i = 0; i < int(cap_lo_info.size()); ++i) result.push_back(get_edge(i));
        return result;
    }
};