//
// Monge 最短路問題に関するアルゴリズム全集
//
// verified
//   AtCoder EDPC Z - Frog 3 (for simple LARSCH, LARSCH, monge CHT)
//     https://atcoder.jp/contests/dp/tasks/dp_z 
//
//   Codeforces Round 189 (Div. 1) C. Kalila and Dimna in the Logging Industry (for simple LARSCH, LARSCH, monge CHT)
//     https://codeforces.com/contest/319/problem/C 
// 
//   yukicoder No.705 ゴミ拾い Hard (for simple LARSCH, LARSCH, monge CHT)
//     https://yukicoder.me/problems/no/705 
//
//   ABC 218 H - Red and Blue Lamps (for Alien DP)
//     https://atcoder.jp/contests/abc218/tasks/abc218_h
//
//   yukicoder No.952 危険な火薬庫 (for SMAWK + D-enum)
//     https://yukicoder.me/problems/no/952
//
//   Codeforces Round 438 (Div. 1 + Div. 2 combined) F. Yet Another Minimization Problem
//     2-solutions (slide access Monotone Minima + D-enum, slide access simple LARSCH + Alien DP)
//     https://codeforces.com/contest/868/problem/F
//


#include <bits/stdc++.h>
using namespace std;


// check whether N x M matrix f is Monge or not, in O(NM)
template<class FUNC> bool is_monge
(int N, int M, const FUNC &f, bool upper_triangular = false, bool output_detail = true) {
    assert(N > 0 && M > 0);
    for (int i = 0; i + 1 < N; i++) {
        for (int j = (upper_triangular ? i+2 : 0); j + 1 < M; j++) {
            auto f00 = f(i, j), f01 = f(i, j + 1), f10 = f(i + 1, j), f11 = f(i + 1, j + 1);
            if (f00 + f11 > f10 + f01) {
                if (output_detail) {
                    cout << "Not Monge! " << endl;
                    cout << "f(" << i << ", " << j << ") = " << f(i, j) << ", ";
                    cout << "f(" << i << ", " << j + 1 << ") = " << f(i, j + 1) << endl;
                    cout << "f(" << i + 1 << ", " << j << ") = " << f(i + 1, j) << ", ";
                    cout << "f(" << i + 1 << ", " << j + 1 << ") = " << f(i + 1, j + 1) << endl;
                }
                return false;
            }
        }
    }
    return true;
}

// check whether N x M matrix f is anti-Monge or not, in O(NM)
template<class FUNC> bool is_anti_monge
(int N, int M, const FUNC &f, bool upper_triangular = false, bool output_detail = true) {
    assert(N > 0 && M > 0);
    for (int i = 0; i + 1 < N; i++) {
        for (int j = (upper_triangular ? i+2 : 0); j + 1 < M; j++) {
            auto f00 = f(i, j), f01 = f(i, j + 1), f10 = f(i + 1, j), f11 = f(i + 1, j + 1);
            if (f00 + f11 < f10 + f01) {
                if (output_detail) {
                    cout << "Not anti-Monge! " << endl;
                    cout << "f(" << i << ", " << j << ") = " << f(i, j) << ", ";
                    cout << "f(" << i << ", " << j + 1 << ") = " << f(i, j + 1) << endl;
                    cout << "f(" << i + 1 << ", " << j << ") = " << f(i + 1, j) << ", ";
                    cout << "f(" << i + 1 << ", " << j + 1 << ") = " << f(i + 1, j + 1) << endl;
                }
                return false;
            }
        }
    }
    return true;
}

// find min_j f(i, j) for all i, by Monotone Minima, O(H + W log H)
// f(i, j) must be monotone (argmin is not decreasing)
// slide access OK
template<class VAL, class FUNC> void MonotoneMinimaRec
(int HL, int HR, int WL, int WR, const FUNC &f, vector<pair<VAL, int>> &res) {
    if (HR - HL <= 0) return;
    int HM = (HL + HR) / 2;
    res[HM].second = WL;
    for (int i = WL; i < WR; i++) {
        VAL val = f(HM, i);
        if (res[HM].first > val) res[HM] = make_pair(val, i);
    }
    MonotoneMinimaRec(HL, HM, WL, res[HM].second + 1, f, res);
    MonotoneMinimaRec(HM + 1, HR, res[HM].second, WR, f, res);
}
template<class VAL, class FUNC> vector<pair<VAL, int>> MonotoneMinima(int H, int W, const FUNC &f) {
    vector<pair<VAL, int>> res(H, make_pair(numeric_limits<VAL>::max() / 2, -1));
    MonotoneMinimaRec(0, H, 0, W, f, res);
    return res;
}

// find shortest path on DAG with monotone cost, by D&D Monotone Minima, O(N (log N)^2)
// vertex: 0, 1, 2, ..., N
// f(i, j) must be monotone (argmin is not decreasing)
// slide access OK
template<class VAL, class FUNC> vector<pair<VAL, int>> DDMonotoneMinima(int N, const FUNC &f) {
    vector<pair<VAL, int>> res(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
    res[0].first = VAL(0);
    auto f2 = [&](int i, int j) -> VAL { return res[j].first + f(j, i); };
    auto rec = [&](auto &&rec, int left, int right) -> void {
        if (right - left <= 1) return;
        int mid = (left + right) / 2;
        rec(rec, left, mid);
        MonotoneMinimaRec(mid, right, left, mid, f2, res);
        rec(rec, mid, right);
    };
    rec(rec, 0, N + 1);
    return res;
}

// find min_j f(i, j) for all i, by Monotone Minima, O(H + W log H)
// f(i, j) must be totally monotone
// slide access NG
template<class VAL, class FUNC> void SMAWKRec
(const vector<int> &X, const vector<int> &Y, const FUNC &f, vector<pair<VAL, int>> &res) {
    if (X.empty()) return;

    // Reduce Step
    vector<int> X2, Y2;
    for (auto y : Y) {
        while (!Y2.empty()) {
            int py = Y2.back(), x = X[(int)Y2.size() - 1];
            if (f(x, y) >= f(x, py)) break;
            Y2.pop_back();
        }
        if (Y2.size() < X.size()) Y2.emplace_back(y);
    }

    // Recurse Step
    for (int i = 1; i < (int)X.size(); i += 2) X2.emplace_back(X[i]);
    SMAWKRec(X2, Y2, f, res);

    // Interpolate Step
    int p = 0;
    for (int i = 0; i < (int)X.size(); i += 2) {
        int lim = (i + 1 < (int)X.size() ? res[X[i + 1]].second : Y.back()), best = Y[p];
        while (Y[p] < lim) {
            p++;
            if (f(X[i], Y[p]) < f(X[i], best)) best = Y[p];
        }
        res[X[i]] = {f(X[i], best), best};
    }
}
template<class VAL, class FUNC> vector<pair<VAL, int>> SMAWK(int H, int W, const FUNC &f) {
    if (H == 0) return {};
    assert(W > 0);
    vector<pair<VAL, int>> res(H, make_pair(numeric_limits<VAL>::max() / 2, -1));
    vector<int> X(H), Y(W);
    for (int i = 0; i < H; i++) X[i] = i;
    for (int j = 0; j < W; j++) Y[j] = j;
    SMAWKRec(X, Y, f, res);
    return res;
}

// find shortest path on DAG with totally monotone cost, by D&D SMAWK, O(N log N)
// vertex: 0, 1, 2, ..., N
// f(i, j) must be totally monotone
template<class VAL, class FUNC> vector<pair<VAL, int>> DDSMAWK(int N, const FUNC &f) {
    vector<pair<VAL, int>> res(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
    vector<pair<VAL, int>> tmp(N + 1);
    res[0].first = VAL(0);
    auto f2 = [&](int i, int j) -> VAL { return res[j].first + f(j, i); };
    auto rec = [&](auto &&rec, int left, int right) -> void {
        if (right - left <= 1) return;
        int mid = (left + right) / 2;
        vector<int> X(right - mid), Y(mid - left);
        for (int i = mid; i < right; i++) X[i - mid] = i;
        for (int j = left; j < mid; j++) Y[j - left] = j;
        rec(rec, left, mid);
        SMAWKRec(X, Y, f2, tmp);
        for (auto x : X) if (tmp[x].first < res[x].first) res[x] = tmp[x];
        rec(rec, mid, right);
    };
    rec(rec, 0, N + 1);
    return res;
}

// noshi's simplified LARSCH
// find shortest path from vertex 0 on DAG in O(N log N)
// vertex: 0, 1, 2, ..., N, f(i, j) must be Monge
// slide access OK
template<class VAL> struct SimpleLARSCH {
    VAL INF = numeric_limits<VAL>::max() / 2;
    int CNT_INF = numeric_limits<int>::max() / 2;

    // results
    vector<VAL> dp;
    vector<int> cnt, prev;

    // solver (random access ver)
    template<class FUNC> vector<pair<VAL, int>> solve(int N, const FUNC &f, bool minimize_cnt = true) {
        dp.assign(N + 1, INF);
        cnt.assign(N + 1, CNT_INF);
        prev.assign(N + 1, 0);
        dp[0] = 0, cnt[0] = 0;

        auto relax = [&](int l, int r) -> void {
            VAL val = dp[l] + f(l, r);
            int cn = cnt[l] + 1;
            if (dp[r] > val || (dp[r] == val && (minimize_cnt ? cn < cnt[r] : cn > cnt[r]))) {
                dp[r] = val;
                cnt[r] = cn;
                prev[r] = l;
            }
        };
        auto rec = [&](auto &&rec, int l, int r) -> void {
            if (r - l <= 1) return;
            int m = (l + r) / 2;
            for (int k = prev[l]; k <= prev[r]; k++) relax(k, m);
            rec(rec, l, m);
            for (int k = l + 1; k <= m; k++) relax(k, r);
            rec(rec, m, r);
        };

        if (N > 0) relax(0, N), rec(rec, 0, N);
        vector<pair<VAL, int>> res(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
        res[0].first = VAL(0);
        for (int i = 1; i <= N; i++) res[i] = {dp[i], prev[i]};
        return res;
    }

    // solver (slide access ver)
    template<class STATE, class ADD, class DEL, class GETCOST> vector<pair<VAL, int>> solve
    (int N, const STATE &ini, const ADD &add, const DEL &del, const GETCOST &get, bool minimize_cnt = true) {
        dp.assign(N + 1, INF);
        cnt.assign(N + 1, CNT_INF);
        prev.assign(N + 1, 0);
        dp[0] = 0, cnt[0] = 0;

        STATE win[2] = { ini, ini };
        int cl[2] = {0, 0}, cr[2] = {0, 0};

        auto move_cursor = [&](int c, int l, int r) {
            while (cr[c] < r) add(win[c], cr[c]++);
            while (cl[c] > l) add(win[c], --cl[c]);
            while (cr[c] > r) del(win[c], --cr[c]);
            while (cl[c] < l) del(win[c], cl[c]++);
        };
        auto relax = [&](int c, int l, int r) {
            move_cursor(c, l, r);
            VAL val = dp[l] + get(win[c]);
            int cn = cnt[l] + 1;
            if (dp[r] > val || (dp[r] == val && (minimize_cnt ? cn < cnt[r] : cn > cnt[r]))) {
                dp[r] = val;
                cnt[r] = cn;
                prev[r] = l;
            }
        };
        auto rec = [&](auto &&rec, int l, int r) -> void {
            if (r - l <= 1) return;
            int m = (l + r) / 2;
            for (int k = prev[l]; k <= prev[r]; k++) relax(0, k, m);
            rec(rec, l, m);
            for (int k = l + 1; k <= m; k++) relax(1, k, r);
            rec(rec, m, r);
        };

        if (N > 0) relax(0, 0, N), rec(rec, 0, N);
        vector<pair<VAL, int>> res(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
        res[0].first = VAL(0);
        for (int i = 1; i <= N; i++) res[i] = {dp[i], prev[i]};
        return res;
    }

    vector<int> reconstruct() {
        int N = (int)dp.size() - 1;
        vector<int> path;
        for (int v = N; v > 0; v = prev[v]) path.emplace_back(v);
        path.emplace_back(0);
        reverse(path.begin(), path.end());
        return path;
    }
};

// LARSCH, in O(N)
// vertex: 0, 1, 2, ..., N, f(i, j) must be Monge
template<class VAL, class FUNC> struct LARSCH {
    struct RowReduce;
    struct ColReduce;
    struct ColMap {
        const ColMap* parent = nullptr;
        const vector<int> *v = nullptr;
        constexpr int mapping(int i) const {
            int x = (v ? (*v)[i] : i);
            return (parent ? parent->mapping(x) : x);
        }
    };
    struct Eval {
        const FUNC *f = nullptr;
        long long a = 1, b = 0;  // row = a * i + b
        const ColMap *cm = nullptr;
        constexpr VAL operator () (int i, int j) const {
            int i2 = int(a * i + b), j2 = (cm ? cm -> mapping(j) : j);
            return (*f)(i2, j2);
        }
    };
    struct RowReduce {
        int N;
        Eval e;
        int cur_row = 0, state = 0;
        unique_ptr<ColReduce> rec;
        constexpr RowReduce(int N_, const Eval &e_) : N(N_), e(e_) {
            int M = N / 2;
            if (M > 0) {
                Eval e2 = e;
                e2.a = e.a * 2, e2.b = e.a + e.b;
                rec = make_unique<ColReduce>(M, e2);
            }
        }
        constexpr void reset() {
            cur_row = 0, state = 0;
            if (rec) rec->reset();
        }
        int get_argmin() {
            int i = cur_row++;
            if (!(i & 1)) {
                int prev = state, next = (i + 1 == N ? N - 1 : rec->get_argmin());
                state = next;
                int res = prev;
                for (int j = prev + 1; j <= next; j++) {
                    if (e(i, res) > e(i, j)) res = j;
                }
                return res;
            } else {
                return (e(i, state) <= e(i, i) ? state : i);
            }
        }
    };
    struct ColReduce {
        int N;
        Eval e;
        int cur_row = 0;
        vector<int> cols;
        ColMap cm;
        RowReduce rec;
        constexpr ColReduce(int N_, const Eval &e_) 
        : N(N_), e(e_), cols(), cm{e.cm, &cols}, rec(N_, Eval{e.f, e.a, e.b, &cm}) {
            cols.reserve(N);
        } 
        constexpr void reset() {
            cur_row = 0;
            cols.clear();
            rec.reset();
        }
        constexpr void push_col(int i, int j) {
            while (!cols.empty()) {
                int siz = (int)cols.size();
                if (siz == i) break;
                int last = cols.back();
                if (e(siz - 1, last) > e(siz - 1, j)) cols.pop_back();
                else break;
            }
            if ((int)cols.size() != N) cols.emplace_back(j);
        }
        constexpr int get_argmin() {
            int i = cur_row++;
            if (i == 0) {
                cols.clear();
                cols.emplace_back(0);
            } else {
                push_col(i, i * 2 - 1);
                push_col(i, i * 2);
            }
            return cols[rec.get_argmin()];
        }
    };

    FUNC f;
    ColMap root_cm;
    Eval root_eval;
    unique_ptr<RowReduce> base;
    explicit LARSCH(int N_, const FUNC &f_) 
    : f(std::move(f_)), root_cm{nullptr, nullptr}, root_eval{&f, 1, 0, &root_cm} {
        base = make_unique<RowReduce>(N_, root_eval);
    }
    constexpr void reset() {
        base->reset();
    }
    constexpr int get_argmin() {
        return base->get_argmin();
    }
};

// Monge Shortest Path Wrapper
template<class VAL> struct MongeShortestPath {
    VAL INF = numeric_limits<VAL>::max() / 2;
    int CNT_INF = numeric_limits<int>::max() / 2;

    // results
    vector<VAL> dp;
    vector<int> cnt, prev;

    // solver (random access ver)
    template<class FUNC> vector<pair<VAL, int>> solve(int N, const FUNC &f
    , const string solver = "simple_larsch") {
        vector<pair<VAL, int>> res(N + 1, make_pair(numeric_limits<VAL>::max() / 2, -1));
        if (solver == "simple_larsch") {
            SimpleLARSCH<VAL> slar;
            res = slar.solve(N, f, true);
            dp = slar.dp, cnt = slar.cnt, prev = slar.prev;
        } else if (solver == "larsch") {
            dp.assign(N + 1, INF);
            cnt.assign(N + 1, CNT_INF);
            prev.assign(N + 1, 0);
            dp[0] = 0, cnt[0] = 0;
            auto f2 = [&](int i, int j) -> VAL {
                i++;
                if (i <= j) return INF;
                else return dp[j] + f(j, i);
            };
            LARSCH<VAL, decltype(f2)> lar(N, f2);
            for (int r = 1; r <= N; r++) {
                int l = lar.get_argmin();
                dp[r] = dp[l] + f(l, r);
                prev[r] = l;
                cnt[r] = cnt[l] + 1;
            }
            res[0].first = VAL(0);
            for (int r = 1; r <= N; r++) res[r] = {dp[r], prev[r]};
        } else if (solver == "monotone") {
            res = DDMonotoneMinima<VAL>(N, f);
        } else if (solver == "smawk") {
            res = DDSMAWK<VAL>(N, f);
        } else {
            cout << "no solver" << endl;
            assert(true);
        }
        return res;
    }

    // solver (slide access ver)
    template<class STATE, class ADD, class DEL, class GETCOST> vector<pair<VAL, int>> solve
    (int N, const STATE &ini, const ADD &add, const DEL &del, const GETCOST &get
    , const string solver = "simple_larsch") {
        vector<pair<VAL, int>> res;
        if (solver == "simple_larsch") {
            SimpleLARSCH<VAL> sl;
            res = sl.solve(N, ini, add, del, get, true);
            dp = sl.dp, cnt = sl.cnt, prev = sl.prev;
        } else {
            cout << "no solver" << endl;
            assert(true);
        }
        return res;
    }

    vector<int> reconstruct() {
        int N = (int)dp.size() - 1;
        vector<int> path;
        for (int v = N; v > 0; v = prev[v]) path.emplace_back(v);
        path.emplace_back(0);
        reverse(path.begin(), path.end());
        return path;
    }
};

// find the d-edges shortest path for d = 1, 2, ..., D, in O(ND)
// vertex: 0, 1, 2, ..., N
// you can choose SMAWK (random access) or Monotone Minima (slide access)
template<class VAL> struct MongeShortestPathWithDEdges {
    VAL INF = numeric_limits<VAL>::max() / 2;

    // solver
    template<class FUNC> vector<VAL> solve(int N, const FUNC &f, int D, const string &solver = "smawk") {
        assert(D <= N);
        vector<VAL> res{VAL(0)}, dp(N + 1, INF);
        dp[0] = 0;
        auto f2 = [&](int i, int j) -> VAL { return (j < i ? dp[j] + f(j, i) : INF); };
        for (int d = 1; d <= D; d++) {
            vector<pair<VAL, int>> tmp;
            if (solver == "smawk") tmp = SMAWK<VAL>(N + 1, N + 1, f2);
            else if (solver == "monotone") tmp = MonotoneMinima<VAL>(N + 1, N + 1, f2);
            for (int i = d; i <= N; i++) dp[i] = tmp[i].first;
            res.emplace_back(dp[N]);
        }
        return res;
    }
};

/*
    Alien's Trick (by ラグランジュ緩和)

    min_{x}: f(x) s.t. g(x) = K
    = max_{λ は整数}: min_{x} (f(x) + λ(g(x) - K))

    max_{x}: f(x) s.t. g(x) = K
    = min_{λ は整数}: max_{x} (f(x) + λ(g(x) - K))

    【仮定】
　　・f(x), g(x): 0 以上の整数値をとる関数
　　・K: 整数
　　・g(x) = p となる x についての f(x) の最小値を h(p) とおくと、h(p) は p について下に凸な関数
　　・h(K) < ∞ (g(x) = K となる x が存在)

    【インターフェース】
　　・lag(λ): x✳︎ = argmin_{x} {f(x) + λ(g(x) - K)} としたときのペア値 {f(x✳︎), g(x✳︎)} を返す関数
　　・返り値: {最適値, そのときの λ}
*/
template<class VAL> struct AliensTrick {
    // find lo such that g(lo) >= R
    template<class FUNC> tuple<VAL, VAL, VAL> find_low(const FUNC &lag, VAL R) {
        auto [f0, g0] = lag(VAL(0));
        VAL lo = 0, hi = 0, flo = f0, glo = g0;
        if (glo >= R) return tie(lo, flo, glo);
        lo = VAL(-1);
        tie(flo, glo) = lag(lo);
        while (glo < R) {
            assert(lo < hi);
            VAL dif = hi - lo;
            hi = lo, lo -= VAL(2) * dif;
            tie(flo, glo) = lag(lo);
        }
        return tie(lo, flo, glo);
    }

    // find hi such that g(hi) <= L
    template<class FUNC> tuple<VAL, VAL, VAL> find_high(const FUNC &lag, VAL L) {
        auto [f0, g0] = lag(VAL(0));
        VAL lo = 0, hi = 0, fhi = f0, ghi = g0;
        if (ghi <= L) return tie(hi, fhi, ghi);
        hi = VAL(1);
        tie(fhi, ghi) = lag(hi);
        while (ghi > L) {
            assert(lo < hi);
            VAL dif = hi - lo;
            lo = hi, hi += VAL(2) * dif;
            tie(fhi, ghi) = lag(hi);
        }
        return tie(hi, fhi, ghi);
    }

    // solver
    template<class FUNC> pair<VAL, VAL> solve(const FUNC &lag, VAL K) {
        // zero check
        auto [f0, g0] = lag(VAL(0));
        if (g0 == K) return {f0, VAL(0)};

        // find an initial value of binary search (by doubling)
        auto [lo, flo, glo] = find_low(lag, K);
        auto [hi, fhi, ghi] = find_high(lag, K);
        if (glo == K) return {flo, lo};
        if (ghi == K) return {fhi, hi};
        assert(glo > K && ghi < K);

        // binary search
        while (hi - lo > VAL(1)) {
            VAL mid = (lo + hi) / 2;
            auto [fmid, gmid] = lag(mid);
            if (gmid == K) return {fmid, mid};
            if (gmid > K) lo = mid, flo = fmid, glo = gmid;
            else hi = mid, fhi = fmid, ghi = gmid;
        }

        // in case: h(p) has a collinear part (= could not achieve g(x) = K)
        // return the Lagrange value
        VAL reslo = flo + lo * (glo - K), reshi = fhi + hi * (ghi - K);
        if (reslo >= reshi) return {reslo, lo};
        else return {reshi, hi};
    }
    template<class FUNC> pair<VAL, VAL> minimize(const FUNC &lag, VAL K) {
        return solve(lag, K);
    }
    template<class FUNC> pair<VAL, VAL> maximize(const FUNC &lag, VAL K) {
        auto reverse_lag = [&](VAL lambda) -> pair<VAL, VAL> {
            auto [f, g] = lag(-lambda);
            return {-f, g};
        };
        auto [f, g] = solve(reverse_lag, K);
        return {-f, -g};
    }

    // in case: L <= g(x) <= R
    template<class FUNC> pair<VAL, VAL> solve(const FUNC &lag, VAL L, VAL R) {
        assert(L <= R);
        auto [f0, g0] = lag(VAL(0));
        if (L <= g0 && g0 <= R) return {f0, VAL(0)};
        if (g0 > R) return solve(lag, R);
        else return solve(lag, L);
    }
    template<class FUNC> pair<VAL, VAL> minimize(const FUNC &lag, VAL L, VAL R) {
        return solve(lag, L, R);
    }
    template<class FUNC> pair<VAL, VAL> maximize(const FUNC &lag, VAL L, VAL R) {
        auto reverse_lag = [&](VAL lambda) -> pair<VAL, VAL> {
            auto [f, g] = lag(-lambda);
            return {-f, g};
        };
        auto [f, g] = solve(reverse_lag, L, R);
        return {-f, -g};
    }
};

// Alien DP
// find the d-edges shortest path from vertex 0 on DAG by Lagrange relaxation
// vertex: 0, 1, 2, ..., N, f(i, j) must be Monge
template<class VAL> struct AlienDP {
    // inner values
    MongeShortestPath<VAL> msp;
    AliensTrick<VAL> at;

    // solver (random access ver.)
    template<class FUNC> pair<VAL, VAL> solve(int N, const FUNC &f, int D
    , const string &solver = "simple_larsch") {
        auto lag = [&](VAL lambda) -> pair<VAL, VAL> {
            auto fg = [&](int i, int j) -> VAL { return f(i, j) + lambda; };
            const auto &obj = msp.solve(N, fg, solver);
            auto cnt = msp.cnt[N];
            auto res = obj.back().first - lambda * cnt;
            return {res, cnt};
        };
        return at.solve(lag, D);
    }

    // solver (slide access ver.)
    template<class STATE, class ADD, class DEL, class GETCOST> pair<VAL, int> solve
    (int N, const STATE &ini, const ADD &add, const DEL &del, const GETCOST &get, int D
    , const string &solver = "simple_larsch") {
        auto lag = [&](VAL lambda) -> pair<VAL, VAL> {
            auto get_lag = [&](STATE &st) -> VAL { return get(st) + lambda; };
            const auto &obj = msp.solve(N, ini, add, del, get_lag, solver);
            auto cnt = msp.cnt[N];
            auto res = obj.back().first - lambda * cnt;
            return {res, cnt};
        };
        return at.solve(lag, D);
    }

    // debugger
    template<class FUNC> void debug(int N, const FUNC &f, VAL start = 0, VAL goal = 10) {
        auto lag = [&](VAL lambda) -> void {
            auto fg = [&](int i, int j) -> VAL { return f(i, j) + lambda; };
            const auto &dp = msp.solve(N, fg);
            auto cnt = msp.cnt[N];
            auto res = dp.back().first - lambda * cnt;
            cout << lambda << ": " << res << "(" << cnt << "); (";
            for (int i = 0; i < (int)dp.size(); i++) {
                if (i) cout << " ";
                cout << "<" << dp[i].first << ", " << dp[i].second << ">";
            }
            cout << ")" << endl;
        };
        for (VAL lambda = start; lambda <= goal; lambda++) lag(lambda);
    }
};

// f: Monge, interval monotone ((N+1) x (N+1))
// dp[i][j] = min_{i <= k < j} (dp[i][k] + dp[k][j]) + f(i, j), in O(N^2)
template<class VAL, class FUNC> VAL KnuthYao(int N, const FUNC &f) {
    VAL INF = numeric_limits<VAL>::max() / 2;
    vector<vector<VAL>> dp(N+1, vector<VAL>(N+1, INF));
    vector<vector<int>> K(N+1, vector<int>(N+1, -1));

    for (int i = 0; i <= N; i++) {
        dp[i][i] = 0, K[i][i] = i;
        if (i < N) dp[i][i+1] = f(i, i+1), K[i][i+1] = i;
    }
    for (int between = 2; between <= N; between++) {
        for (int i = 0; i + between <= N; i++) {
            int j = i + between;
            for (int k = K[i][j-1]; k <= K[i+1][j]; k++) {
                VAL tmp = dp[i][k] + dp[k][j] + f(i, j);
                if (dp[i][j] > tmp) {
                    dp[i][j] = tmp;
                    K[i][j] = k;
                }   
            }
        }
    }
    return dp[0][N];
}

// Convex Hull Trick (using Li Chao Tree)
/*
    Func：Monge 性を仮定
    - MIN: クエリ x の取りうる最小値
    - MAX: クエリ x の取りうる最大値 (f_i(MAX) がオーバーフローしないように注意）
    - INF: 最大値
 
    - insert (Func f_i): add f_i, O(log N)
    - query (x): min_i{ f_i(x) }, O(log N)
*/
template<class T> struct CHT {
    using Func = function<T(T)>;
    struct Node {
        Func func;
        Node *left, *right;
        Node(const Func& f) : left(nullptr), right(nullptr) {
            func = f;
        }
    };
    
    const T MIN, MAX, INF;
    Node* root;
    
    CHT(T MIN, T MAX, T INF) : MIN(MIN), MAX(MAX), INF(INF), root(nullptr) { }
    Node* insert(Node* p, T low, T high, Func& f) {
        if (!p) return new Node(f);
        if (p->func(low) <= f(low) && p->func(high) <= f(high)) return p;
        if (p->func(low) >= f(low) && p->func(high) >= f(high)) {
            p->func = f;
            return p;
        }
        T mid = (low + high) / 2;
        if (p->func(mid) > f(mid)) swap(p->func, f);
        if (p->func(low) >= f(low)) p->left = insert(p->left, low, mid, f);
        else p->right = insert(p->right, mid, high, f);
        return p;      
    }
    void insert(Func f) {
        root = insert(root, MIN, MAX, f);
    }
    T query(Node* p, T low, T high, T x) {
        if (!p) return INF;
        if (low == high) return p->func(x);
        T mid = (low + high) / 2;
        if (x <= mid) return min(p->func(x), query(p->left, low, mid, x));
        else return min(p->func(x), query(p->right, mid, high, x));
    }
    T query(T x) {
        return query(root, MIN, MAX, x);
    }
};


//------------------------------//
// Examples
//------------------------------//

// AtCoder EDPC Z - Frog 3
/*
    H は単調増加数列
    chmin(dp[j], dp[i] + (H[j] - H[i])^2 + C)
    i -> j のコスト：(H[j] - H[i])^2 ...... 差の凸関数は Monge
    スタート: 0, ゴール: N-1
*/
void EDPC_Z(const string solver = "simple_larsch") {
    long long N, C;
    cin >> N >> C;
    vector<long long> H(N);
    for (long long i = 0; i < N; i++) cin >> H[i];
    auto func = [&](int i, int j) -> long long {
        return (H[j] - H[i]) * (H[j] - H[i]) + C;
    };
    MongeShortestPath<long long> msp;
    const auto &res = msp.solve(N-1, func, solver);
    cout << res[N-1].first << endl;
}

// Codeforces Round 189 (Div. 1) C. Kalila and Dimna in the Logging Industry
/*
    A: 単調増加, B: 単調減少, ともに長さ N
    i -> j のコストが、B[i] × A[j] で与えられる ..... 単調増加 × 単調減少は Monge
    スタート: 0, ゴール: N-1
*/
void Codeforces_189_C(const string solver = "simple_larsch") {
    long long N;
    cin >> N;
    vector<long long> A(N), B(N);
    for (int i = 0; i < N; i++) cin >> A[i];
    for (int i = 0; i < N; i++) cin >> B[i];
    auto func = [&](int i, int j) -> long long {
        return B[i] * A[j];
    };
    MongeShortestPath<long long> msp;
    auto res = msp.solve(N-1, func, solver);
    cout << res[N-1].first << endl;
}

// yukicoder No.705 ゴミ拾い Hard
/*
    A, X, Y: N 個
    これらを区間に分割していく
    　dp[j] = min_{0 ≦ i < j}(dp[i] + |A[j-1] - X[i]|^3 + |-Y[i]|^3)
    i -> j のコスト：|A[j-1] - X[i]|^3 + |-Y[i]|^3 ...... 差の凸関数 (Monge) + 縞々 (Monge) -> Monge
    スタート: 0, ゴール: N
*/
void yukicoder_705(const string solver = "simple_larsch") {
    int N;
    cin >> N;
    vector<long long> A(N), X(N), Y(N);
    for (int i = 0; i < N; i++) cin >> A[i];
    for (int i = 0; i < N; i++) cin >> X[i];
    for (int i = 0; i < N; i++) cin >> Y[i];
    auto func = [&](int i, int j) -> long long {
        long long dx = abs(A[j-1] - X[i]), dy = abs(Y[i]);
        return dx * dx * dx + dy * dy * dy;
    };
    MongeShortestPath<long long> msp;
    auto res = msp.solve(N, func, solver);
    cout << res[N].first << endl;
}

// ABC 218 H - Red and Blue Lamps
void ABC_218_H() {
    long long N, R;
    cin >> N >> R;
    vector<long long> A(N - 1);
    for (int i = 0; i < N - 1; i++) cin >> A[i];
    auto cost = [&](int i, int j) -> long long {
        if (j - i <= 1) return 0;
        if (i == 0) return -A[j - 2];
        else return -A[i - 1] - A[j - 2];
    };
    AlienDP<long long> ad;
    //ad.debug(N, cost);
    auto [resR, cntR] = ad.solve(N, cost, R);
    auto [resB, cntB] = ad.solve(N, cost, N - R);
    auto res = max(-resR, -resB);
    cout << res << endl;
}

// yukicoder No.952 危険な火薬庫
/*
    末尾にドアを追加する。
    ノード 0, 1, ..., N, N+1
    f(i, j) = (S[j-1] - S[i])^2
    選ぶドアが k 個 → 区間の個数は、N+1-k
*/
void yukicoder_952() {
    int N;
    cin >> N;
    vector<long long> A(N + 1, 0), S(N + 2, 0);
    for (int i = 0; i < N; i++) cin >> A[i], S[i + 1] = S[i] + A[i];
    auto f = [&](int i, int j) -> long long { 
        return (S[j - 1] - S[i]) * (S[j - 1] - S[i]);
    };
    MongeShortestPathWithDEdges<long long> msp;
    auto res = msp.solve(N + 1, f, N + 1);
    for (int k = 1; k <= N; k++) cout << res[N + 1 - k] << endl;
}

// Codeforces Round 438 (Div. 1 + Div. 2 combined) F. Yet Another Minimization Problem
// by Monge 単一始点 d-辺最短路の d = 1, 2, ..., D における列挙 (by SMAWK 法, in O(ND))
void Codeforces438_F_enum() {
    long long N, K;
    cin >> N >> K;
    vector<long long> A(N);
    for (int i = 0; i < N; i++) cin >> A[i];

    vector<long long> nums(N + 1, 0);
    long long left = 0, right = 0, score = 0;
    auto add = [&](int i) -> void {
        score += nums[A[i]];
        nums[A[i]]++;
    };
    auto del = [&](int i) -> void {
        nums[A[i]]--;
        score -= nums[A[i]];
    };
    auto f = [&](int i, int j) -> long long {
        while (left < i) del(left++);
        while (i < left) add(--left);
        while (right < j) add(right++);
        while (j < right) del(--right);
        return score;
    };
    MongeShortestPathWithDEdges<long long> msp;
    auto res = msp.solve(N, f, K, "monotone");  // we should use slide access
    cout << res[K] << endl;
}

// Codeforces Round 438 (Div. 1 + Div. 2 combined) F. Yet Another Minimization Problem
// by Alien DP (not the assumed solution)
void Codeforces438_F_alien() {
    long long N, K;
    cin >> N >> K;
    vector<long long> A(N);
    for (int i = 0; i < N; i++) cin >> A[i];

    struct STATE {
        vector<long long> nums;
        long long cost;
        STATE(int N) : nums(N + 1, 0), cost(0) {}
    };
    STATE ini(N);
    auto add = [&](STATE &st, int i) -> void {
        st.cost += st.nums[A[i]];
        st.nums[A[i]]++;
    };
    auto del = [&](STATE &st, int i) -> void {
        st.nums[A[i]]--;
        st.cost -= st.nums[A[i]];
    };
    auto get = [&](STATE &st) -> long long {
        return st.cost;
    };

    AlienDP<long long> adp;
    auto res = adp.solve(N, ini, add, del, get, K);  // we should use slide access
    cout << res.first << endl;
}


int main() {
    //EDPC_Z("simple_larsch");
    //EDPC_Z("larsch");
    //Codeforces_189_C("simple_larsch");
    //Codeforces_189_C("larsch");
    yukicoder_705("simple_larsch");
    //ABC_218_H();
    //yukicoder_952();
    //Codeforces438_F_enum();
    //Codeforces438_F_alien();
}