#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
// const ll MOD = 1000000007;
const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-12;

struct mint {
    ll x;
    mint(ll x = 0): x((x % MOD + MOD) % MOD){}
    mint operator-() const {return mint(-x);}

    mint& operator+=(const mint a){
        if ((x += a.x) >= MOD) x -= MOD;
        return *this;
    }

    mint& operator-=(const mint a){
        if ((x += MOD - a.x) >= MOD) x -= MOD;
        return *this;
    }

    mint& operator*=(const mint a) {(x *= a.x) %= MOD; return *this;}
    mint operator+(const mint a) const {return mint(*this) += a;}
    mint operator-(const mint a) const {return mint(*this) -= a;}
    mint operator*(const mint a) const {return mint(*this) *= a;}
};
istream& operator>>(istream& is, const mint& a) {return is >> a.x;}
ostream& operator<<(ostream& os, const mint& a) {return os << a.x;}

using edge = struct {ll to; ll cost;};

struct fpoint {ld x; ld y;
    bool operator<(const fpoint& p) const{
        if (x == p.x) return y < p.y;
        return x < p.x;
    }

    bool operator==(const fpoint& p) const{
        if (p.x - EPS < x && x < p.x + EPS
            && p.y - EPS < y && y < p.y + EPS) return true;
        return false;
    }

    bool operator!=(const fpoint& p) const{
        if (p.x - EPS > x || x > p.x + EPS
            || p.y - EPS > y || y > p.y + EPS) return true;
        return false;
    }

    fpoint& operator+=(const ld a){
        x += a, y += a;
        return *this;
    }

    fpoint& operator-=(const ld a){
        x -= a, y -= a;
        return *this;
    }

    fpoint& operator*=(const ld a){
        x *= a, y *= a;
        return *this;
    }

    fpoint& operator/=(const ld a){
        x /= a, y /= a;
        return *this;
    }

    fpoint& operator+=(const fpoint p){
        x += p.x, y += p.y;
        return *this;
    }

    fpoint& operator-=(const fpoint p){
        x -= p.x, y -= p.y;
        return *this;
    }

    fpoint& operator*=(const fpoint p){
        x *= p.x, y *= p.y;
        return *this;
    }

    fpoint& operator/=(const fpoint p){
        x /= p.x, y /= p.y;
        return *this;
    }

    bool is_zero(){
        fpoint z = {0, 0};
        return z == *this;
    }

    fpoint operator+(const fpoint p) const {return fpoint(*this) += p;}
    fpoint operator-(const fpoint p) const {return fpoint(*this) -= p;}
    fpoint operator*(const fpoint p) const {return fpoint(*this) *= p;}
    fpoint operator/(const fpoint p) const {return fpoint(*this) /= p;}
    fpoint operator+(const ld a) const {return fpoint(*this) += a;}
    fpoint operator-(const ld a) const {return fpoint(*this) -= a;}
    fpoint operator*(const ld a) const {return fpoint(*this) *= a;}
    fpoint operator/(const ld a) const {return fpoint(*this) /= a;}
};

struct point {ll x; ll y;
    bool operator<(const point& p) const{
        if (x == p.x) return y < p.y;
        return x < p.x;
    }

    bool operator==(const point& p) const{
        if (x == p.x && y == p.y) return true;
        return false;
    }

    bool operator!=(const point& p) const{
        if (x != p.x || y != p.y) return true;
        return false;
    }

    point& operator+=(const ll a){
        x += a, y += a;
        return *this;
    }

    point& operator-=(const ll a){
        x -= a, y -= a;
        return *this;
    }

    point& operator*=(const ll a){
        x *= a, y *= a;
        return *this;
    }

    point& operator+=(const point p){
        x += p.x, y += p.y;
        return *this;
    }

    point& operator-=(const point p){
        x -= p.x, y -= p.y;
        return *this;
    }

    point& operator*=(const point p){
        x *= p.x, y *= p.y;
        return *this;
    }

    bool is_zero(){
        point z = {0, 0};
        return z == *this;
    }

    fpoint to_fpoint(){
        fpoint ret = {ld(x), ld(y)};
        return ret;
    }

    void operator++(int){x++, y++;}
    void operator++(){x++, y++;}
    void operator--(int){x--, y--;}
    void operator--(){x--, y--;}
    point operator+(const point p) const {return point(*this) += p;}
    point operator-(const point p) const {return point(*this) -= p;}
    point operator*(const point p) const {return point(*this) *= p;}
    point operator+(const ll a) const {return point(*this) += a;}
    point operator-(const ll a) const {return point(*this) -= a;}
    point operator*(const ll a) const {return point(*this) *= a;}

};

struct undirected_edge {
    ll from; ll to; ll cost;
    bool operator<(const undirected_edge& ue) const{
        return cost < ue.cost;
    }
};

typedef string str;
typedef std::pair<ll,ll> pl;
typedef std::tuple<ll, ll, ll> tp3;
typedef std::tuple<ll, ll, ll, ll> tp4;
typedef std::map<string, ll> msl;
typedef std::map<char, ll> mcl;
typedef std::map<ll, ll> mll;
typedef std::vector<ll> vl;
typedef std::vector<vl> vl2;
typedef std::vector<vl2> vl3;
typedef std::vector<vl3> vl4;
typedef std::vector<mint> vmi;
typedef std::vector<vmi> vmi2;
typedef std::vector<vmi2> vmi3;
typedef std::vector<vmi3> vmi4;
typedef std::vector<bool> vb;
typedef std::vector<vb> vb2;
typedef std::vector<vb2> vb3;
typedef std::vector<vb3> vb4;
typedef std::vector<pl> vpl;
typedef std::vector<tp3> vtp3;
typedef std::vector<tp4> vtp4;
typedef std::vector<point> points;
typedef std::vector<fpoint> fpoints;
// priority queue. Taking from the higher value. Don't forget calling !q.empty()
typedef std::priority_queue<ll> pq;
// priority queue. Taking from the lower value
typedef std::priority_queue<ll, vl, greater<ll>> pql;
typedef std::vector<vector<edge>> Graph;
const ll N_DIGITS = 60;
const long double PI = 3.14159265358979323846;
points dirs = {{-1, 0}, {1, 0}, {0, 1}, {0, -1},  // four directions
               {1, 1}, {-1, 1}, {-1, -1}, {1, -1},  // diagonal
               {0, 0}  // self
               };

template <typename A, typename B>
string to_string(pair<A, B> p);

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);

string to_string(const string& s) {
  return '"' + s + '"';
}

string to_string(const char* s) {
  return to_string((string) s);
}

string to_string(bool b) {
  return (b ? "true" : "false");
}

string to_string(mint a) {
  return to_string(a.x);
}

string to_string(point p){
  return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";
}

string to_string(fpoint p){
  return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";
}

string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (int i = 0; i < static_cast<int>(v.size()); i++) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}

template <size_t N>
string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) {
    res += static_cast<char>('0' + v[i]);
  }
  return res;
}

template <typename A>
string to_string(A v) {
  bool first = true;
  string res = "{";
  for (const auto &x : v) {
    if (!first) {
      res += ", ";
    }
    first = false;
    res += to_string(x);
  }
  res += "}";
  return res;
}

template <typename A, typename B>
string to_string(pair<A, B> p) {
  return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";
}

template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";
}

template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p) {
  return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";
}

void debug_out() { cerr << endl; }

template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T) {
  cerr << " " << to_string(H);
  debug_out(T...);
}

#define rep(i, n) for(ll (i) = 0; (i) < (n) ; (i)++ )
#define revrep(i, n) for(ll (i) = n - 1; (i) >= 0 ; (i)--)
#define For(i, a, b) for(ll (i) = (a); (i) < (b) ; (i)++ )
#define revFor(i, b, a) for(ll (i) = (b) - 1; (i) >= (a) ; (i)--)
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define isUpper(c) ('a' - c > 0)
#define isNum(c) (0 <= (c) - '0' && (c) - '0' <= 9)
#define toLower(c) char((c)+0x20)
#define toUpper(c) char((c)-0x20)
#define pb push_back
#define mp make_pair
#define mt make_tuple
#define pr(a) std::cout << (a)
#define prl(a) std::cout << (a) << endl
#define prl2(a,b) std::cout << (a) << " " << (b) << endl
#define prl3(a,b,c) std::cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a,b,c,d) std::cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) std::cout << (a) << " "
#define prs2(a,b) std::cout << (a) << " " << (b) << " "
#define prs3(a,b,c) std::cout << (a) << " " << (b) << " " << (c) << " "
#define prs4(a,b,c,d) std::cout << (a) << " " << (b) << " " << (c) << " " << (d) << " "
#define yn(condition) if ((condition)) prl("Yes"); else prl("No");
#define YN(condition) if ((condition)) prl("YES"); else prl("NO");
#define in1(a) cin >> (a)
#define in2(a,b) cin >> (a) >> (b)
#define in3(a,b,c) cin >> (a) >> (b) >> (c)
#define in4(a,b,c,d) cin >> (a) >> (b) >> (c) >> (d)
#define in5(a,b,c,d,e) cin >> (a) >> (b) >> (c) >> (d) >> (e)
#define in6(a,b,c,d,e,f) cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f)
#define in7(a,b,c,d,e,f,g) cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f) >> (g)
#define e1 first
#define e2 second
#define ltos(n) to_string((n))
#define Forchar(c,a,z) for(char (c) = (a); (c) <= (z); (c)++)
#define cntchar(s,c) count(all((s)), c)
#define substring(s, start, end) s.substr((start), (end) - (start) + 1)
#define prl_nd(num, digits) std::cout << fixed << setprecision(digits) << (num) << endl;
#define prl_time(s) prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))

#ifdef _LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif

template<class T>
inline bool chmax(T &a, T b) {
    if(a < b) {a = b; return true;}
    return false;
}

template<class T>
inline bool chmin(T &a, T b) {
    if(a > b){a = b; return true;}
    return false;
}

struct MaxFlow {
    struct F_edge {
        ll to, rev, capacity;
        F_edge(ll to, ll rev, ll capacity) : to(to), rev(rev), capacity(capacity) {}
    };

    typedef vector<F_edge> F_edges;
    vector<F_edges> graph;
    ll n_vertex;
    // level is the shortest path to get a given node from the source node.
    vl level, iter;

    MaxFlow(ll n_vertex) : n_vertex(n_vertex) {graph.resize(n_vertex);}

    void add_edge(ll from, ll to, ll capacity) {
        graph[from].pb({to, ll(graph[to].size()), capacity});
        graph[to].pb({from, ll(graph[from].size()) - 1, 0});
    }

    void bfs(ll source){
        level = vl(n_vertex, -1); level[source] = 0;
        queue<ll> q; q.push(source);
        while (!q.empty()) {
            ll vertex = q.front(); q.pop();
            rep(i, graph[vertex].size()){
                ll target = graph[vertex][i].to;
                ll cap_target = graph[vertex][i].capacity;
                // if the flow can be into the target node, implement below.
                if (cap_target > 0 && level[target] < 0){
                    level[target] = level[vertex] + 1;
                    q.push(target);
                }
            }
        }
    }

    ll dfs(ll vertex, ll sink, ll flow) {
        if (vertex == sink) return flow;
        for(ll &i = iter[vertex]; i < graph[vertex].size(); i++){
            ll target = graph[vertex][i].to; ll cap_target = graph[vertex][i].capacity;
            ll rev_target = graph[vertex][i].rev;
            // if capasitiy is not full yet and target is farther,
            // then assign current flow.
            if (cap_target > 0 && level[vertex] < level[target]) {
                ll d = dfs(target, sink, min(cap_target, flow));
                if (d > 0){ // if the flow successfully reaches the sink, reduce the flow from the capacity 
                    graph[vertex][i].capacity -= d;
                    graph[target][rev_target].capacity += d;
                    return d;
                }
            }
        }
        return 0;
    }

    ll dinic(ll source, ll sink){
        // complexity O(EV^2)
        ll flow = 0;
        while (true) {
            bfs(source);
            // if there is no path leading to the sink, the maximum flow is 0.
            if (level[sink] < 0) return flow;
            iter = vl(n_vertex, 0); ll f;
            while ((f = dfs(source, sink, INF)) > 0) flow += f;
        }
    }
};

struct UnionFind {
    vl parents, set_size;
    set<ll> root_idx;
    ll n_groups;

    UnionFind(ll n){
        parents = set_size = vl(n);
        n_groups = n;
        rep(i, n){
            parents[i] = i;
            set_size[i] = 1LL;
            root_idx.insert(i);
        }
    }

    ll root_find(ll x){
        if (parents[x] == x) return x;
        return parents[x] = root_find(parents[x]);
    }

    void unite(ll x, ll y){
        // priority for x is larger than that of y
        x = root_find(x); y = root_find(y);
        if (x == y) return;
        parents[y] = x, set_size[x] += set_size[y];
        root_idx.erase(y);
        n_groups--;
    }

    bool is_same(ll x, ll y) {// connected or not
        return root_find(x) == root_find(y);
    }

    ll size(ll x) {
        return set_size[root_find(x)];
    }

    ll num_union() const{
        return n_groups;
    }
};

struct Doubling { // ABC167D
    ll n;
    ll sz;
    vl2 next;
    /*
        next[k + 1][i] := next[k][next[k][i]]
        next[0][i] := edge[i]
        e.g. a0, a1, ..., an-1 / 0 <= ai <= n - 1
        a0 -> a[a0] -> a[a[a0]] -> ... -> a[a[...[a[0]]]] (m times)
        Let the function repeatedly input a[i] m times be f[m](a[i])

        - get(i, x) returns f[x](a[i])
        - lower_bound(i, j) returns minimum x which satisfies f[x](a[i]) >= j.
                                    if not possible returns n.

    */

    // edge[i]: the step size for one iteration
    Doubling(vl& edge) : n(edge.size()), sz(62) {
        next.resize(sz, vl(n, -1));
        rep(i, n) next[0][i] = edge[i];
        rep(k, sz - 1) rep(i, n) next[k + 1][i] = next[k][next[k][i]];
    }

    ll get(ll i, ll x) {
        ll ret = i;
        rep(bit, sz){
            if (!(x >> bit & 1)) continue;
            ret = next[bit][ret];
        }
        return ret;
    }

    ll lower_bound(ll i, ll j) {
        ll cur = i, acc = 0;
        revrep(wid, sz){
            if (next[wid][cur] < j) {
                acc += 1LL << wid;
                cur = next[wid][cur];
            }
        }
        return min(n, acc + 1);
    }
};

class LowestCommonAncestor{
public:
    ll N, logN;
    vl depth, len;
    Graph tree;
    vl2 parents;

    LowestCommonAncestor(ll n, ll offset = -1, bool is_weighted = true){
        N = n;
        logN = 0;
        while (N > (1LL << logN)) logN++;
        depth = len = vl(N);
        parents = vl2(logN, vl(N));
        tree = Graph(N);
        input(N, offset, is_weighted);
        init(0, -1, 0, 0);
        build();
    }

    void add_edge(ll from, ll to, ll dist){
        tree[from].pb({to, dist});
        tree[to].pb({from, dist});
    }

    void input(ll n, ll offset = -1, bool is_weighted = true){
        rep(i, n - 1){
            ll a, b, d = 1; in2(a, b);
            a += offset, b += offset;
            if (is_weighted) in1(d);
            add_edge(a, b, d);
        }
    }

    void init(ll source, ll parent, ll d, ll l){
        depth[source] = d;
        parents[0][source] = parent;
        len[source] = l;
        rep(i, tree[source].size()){
            ll target = tree[source][i].to;
            ll cost = tree[source][i].cost;
            if (target == parent) continue;
            init(target, source, d + 1, cost + l);
        }
    }

    void build(){
        rep(k, logN - 1) rep(n, N){
            // if there is no parent, -1.
            // otherwise, the parent of the parent is the parent.
            if (parents[k][n] < 0) parents[k + 1][n] = -1;
            else parents[k + 1][n] = parents[k][parents[k][n]];
        }
    }

    ll query(ll u, ll v){
        if (depth[u] > depth[v]) swap(u, v);
        rep(k, logN) if ((depth[v] - depth[u]) >> k & 1) v = parents[k][v];
        if (u == v) return u;

        revrep(k, logN){
            if (parents[k][u] != parents[k][v]){
                u = parents[k][u]; v = parents[k][v];
            }
        }
        return parents[0][u];
    }

    ll distance(ll u, ll v){
        ll w = query(u, v);
        return len[u] + len[v] - 2 * len[w];
    }

};

struct BinaryIndexedTree{
    ll n, ini; vl dat;
    BinaryIndexedTree(ll n, ll ini = 0):
        dat(n + 1, ini), n(n), ini(ini){};

    //       x: 1001 1010  1100 1011 1101  1111
    // x & - x: 0001 0010  0100 0001 0001  0001
    //      ->: 1010 1100 10000 1100 1100 10000

    ll update_func(ll val, ll d){
        // if maximum -> max(val, dat)
        // return max(val, d);
        // if cumulative sum
        return val + d;
    }

    ll query(ll i){
        /*
        v[0] + v[1] + ... + v[i]
        e.g.) i = 10101
        itr1. 10101 -> 10100
        itr2. 10100 -> 10000
        itr3. 10000 -> 00000 (break)
        */

        if (i < 0) return ini;
        ll ret = 0;
        for (ll j = i; j >= 0; j = (j & (j + 1)) - 1){
            ret = update_func(ret, dat[j]);
        }
        return ret;
    }

    ll query(ll l, ll r){
        // a[l] + a[l + 1] + ... + a[r - 1] + a[r]
        return query(r) - query(l - 1);
    }

    ll lower_bound(ll key){
        //v[0] + v[1] + ... + v[left - 1] < key <= v[0] + v[1] + ... + v[left]
        if (key <= 0) return 0;
        ll left = 0, right = 1;
        while(right <= n) right *= 2;

        for (ll i = right; i > 0; i /= 2){
            if (left + i <= n && dat[left + i - 1] < key){
                key -= dat[left + i - 1];
                left += i;
            }
        }
        return left;
    }

    void update(ll i, ll val){
        /*
        e.g.) i = 10101, n = 11111
        itr1. i: 10101, i+1: 10110 -> 10111
        itr2. i: 10111, i+1: 11000 -> 11111 (break)
        */
        if (i < 0) return;
        for(ll j = i; j < n; j |= j + 1){
            dat[j] = update_func(val, dat[j]);
        }
    }
};

struct SegmentTree{
	ll n, ini, minimize;
	vl dat;

    // when seeking minimum
    // ini = INF
    // when seeking maximum
    // ini = -INF
	SegmentTree(ll n_, bool minimize_ = true){
        n = 1; minimize = minimize_;
        if (minimize) ini = INF;
        else ini = - INF;
		while (n < n_) n *= 2;
		dat.resize(2 * n - 1);
		rep(i, 2 * n - 1) dat[i] = ini;
    };

	void update(ll idx, ll val){
		idx += n - 1;
        if (minimize && dat[idx] <= val) return;
        if (!minimize && dat[idx] >= val) return;

		dat[idx] = val;
		while (idx > 0){
			idx = (idx - 1) / 2;
            // when seeking minimum
			if (minimize) dat[idx] = min(dat[idx * 2 + 1], dat[idx * 2 + 2]);
            // when seeking maximum
            else dat[idx] = max(dat[idx * 2 + 1], dat[idx * 2 + 2]);
		}
	}

	ll query(ll l, ll r){
        // ### NOTE ###
        // the range is [l, r]
        // l, l + 1, ..., r
        r++; // to adjust to this method
        return query_segment(l, r, 0, 0, n);
	}

	ll query_segment(ll a, ll b, ll idx, ll l, ll r){
        assert(a < b);

		if (r <= a || b <= l) return ini;
		if (a <= l && r <= b) return dat[idx];
		else{
            ll seg1 = query_segment(a, b, idx * 2 + 1, l, (l + r) / 2);
            ll seg2 = query_segment(a, b, idx * 2 + 2, (l + r) / 2, r);
            // when seeking minimum
            if (minimize) return min(seg1, seg2);
            // when seeking maximum
            else return max(seg1, seg2);
        }
	}
};

template<class Target>
class RerootingTreeDP {
public:
    using T = typename Target::type;
    struct DP_edge {
        ll to, rev; // rev is the index to trace the source node.
        T value; // objective value
    };
private:
    ll n;
    void dfs_fwd(ll source, ll parent){
        ll par_idx = -1;
        vector<T> values;
        rep(i, tree[source].size()) {
            const DP_edge& e = tree[source][i];
            if (e.to == parent){par_idx = i; continue;}
            dfs_fwd(e.to, source);
            values.pb(e.value);
        }

        // If the parent != -1, update the value on edge from parent to source
        if (par_idx != -1){
            ll src_idx = tree[source][par_idx].rev;
            // update values on the edge from parent to source
            tree[parent][src_idx].value = Target::merge(values);
        }
    }

    void dfs_bwd(ll source, ll parent){
        vector<T> values;
        for (auto&& e: tree[source]) values.pb(e.value);
        values = Target::evaluate(values);

        rep(i, tree[source].size()) {
            const DP_edge& e = tree[source][i];
            if (e.to == parent) continue;
            // tree[e.to][e.rev]: e.to -> source
            tree[e.to][e.rev].value = values[i];
            dfs_bwd(e.to, source);
        }
    }
public:
    UnionFind uf;
    vector<vector<DP_edge>> tree;
    RerootingTreeDP(ll n) : n(n), uf(n), tree(n) {}

    void add_edge(ll u, ll v, T val){
        assert(!uf.is_same(u, v));
        tree[u].pb({v, ll(tree[v].size()), val});
        tree[v].pb({u, ll(tree[u].size()) - 1, val});
        uf.unite(u, v);
    }

    void dp() {
        vb visited(n, false);
        rep(i, n){
            if (visited[uf.root_find(i)]) continue;
            dfs_fwd(i, -1);
            visited[uf.root_find(i)] = true;
        }

        visited.assign(n, false);
        rep(i, n){
            if (visited[uf.root_find(i)]) continue;
            dfs_bwd(i, -1);
            visited[uf.root_find(i)] = true;
        }
    }

    ll size() const {return tree.size();}
};

// ABC160F is one example
// Modify the functions evaluate and merge based on given problems
struct Merger {
    using type = ll;

    // This is the exaple of the number of children
    static type merge(const vector<type>& value){
        // merge the result below the source node
        // each value is from each child node of the source node
        // value[i] := f(child i)
        // Here, we would like to obtain f(source) using f(child i) (i = 0, 1, ..., n_children)
        ll ret = 1;
        for (auto&& v: value) ret += v;
        return ret;
    }

    static vector<type> evaluate(const vector<type>& value){
        // value[i] := f(source -> child i)
        // we would like to obtain f(child i -> source)
        // child j (j != i) is the grandchildren of child i
        // represent f(child i -> source) using f(source -> j) (j != i)
        // L[i + 1] := the result using f(source -> k) (k = 0, 1, ..., i)
        // R[i] := the result using f(source -> k) (k = i, i + 1, ..., n_children)

        const ll n_children = value.size();
        vl L(n_children + 1, 0), R(n_children + 1, 0);
        rep(i, n_children)
            L[i + 1] = L[i] + value[i];
        revrep(i, n_children)
            R[i] = R[i + 1] + value[i];

        vl ret(n_children);
        rep(i, n_children) ret[i] = L[i] + R[i + 1] + 1;

        return ret;
    }
};

struct StronglyConnectedComponents{
    ll n, n_cmp;
    vl2 graph, graph_rev, dag, cmp;
    vl order, visited, cmp_idx;

    StronglyConnectedComponents(){}
    StronglyConnectedComponents(ll sz):
        n(sz), graph(sz), graph_rev(sz),
        visited(sz), cmp_idx(sz){}

    void add_edge(ll from, ll to){
        graph[from].pb(to);
        graph_rev[to].pb(from);
    }

    void input(ll m, ll offset = -1){
        ll a, b;
        rep(i, m){
            in2(a, b);
            add_edge(a + offset, b + offset);
        }
    }

    ll operator[](ll k){
        return cmp_idx[k];
    }

    void dfs_fwd(ll source){
        visited[source] = 1;
        rep(i, graph[source].size()){
            ll target = graph[source][i];
            if (!visited[target]) dfs_fwd(target);
        }
        order.pb(source);
    }

    void dfs_bwd(ll source, ll num){
        visited[source] = 1, cmp_idx[source] = num;

        cmp[num].pb(source);
        rep(i, graph_rev[source].size()){
            ll target = graph_rev[source][i];
            if (!visited[target]) dfs_bwd(target, num);
        }
    }

    ll build(){
        fill(all(visited), 0);
        order.clear();
        rep(i, n)
            if (!visited[i]) dfs_fwd(i);

        fill(all(visited), 0);
        ll num = 0;
        revrep(i, order.size()){
            if (!visited[order[i]]){
                dag.pb(vl());
                cmp.pb(vl());
                dfs_bwd(order[i], num++);
            }
        }

        rep(i, n) for (ll to: graph[i])
            if (cmp_idx[i] != cmp_idx[to])
                dag[cmp_idx[i]].pb(cmp_idx[to]);

        rep(i, num){
            sort(all(dag[i]));
            dag[i].erase(unique(all(dag[i])), dag[i].end());
        }
        return n_cmp = num;
    }
};

struct CombinationMemo{
    ll sz, mod;
    vl facts, facts_inv, minv;

    CombinationMemo(ll sz, ll _mod):
        sz(sz), mod(_mod){
            facts.resize(sz + 5);
            facts_inv.resize(sz + 5);
            minv.resize(sz + 5);
            init();
        }

    void init(){
        facts[0] = facts[1] = 1;
        minv[1] = 1;
        facts_inv[0] = facts_inv[1] = 1;

        For(i, 2, sz + 3){
            facts[i] = (i * facts[i - 1]) % mod;
            minv[i] = mod - minv[mod % i] * (mod / i) % mod;
            facts_inv[i] = facts_inv[i - 1] * minv[i] % mod;
        }
    }

    ll nCk(ll n, ll r){
        if (n == r && n == 0) return 1;
        else if (n <= 0 || r < 0 || r > n) return 0;
        ll val = (facts[n] * facts_inv[n - r]) % mod;
        val *= facts_inv[r];
        return val % mod;
    }

    ll nPk(ll n, ll r){
        if (n == r && n == 0) return 1;
        else if (n <= 0 || r < 0 || r > n) return 0;
        ll val = (facts[n] * facts_inv[n - r]) % mod;
        return val % mod;
    }
};

struct PowerMemo{
    ll sz, mod, base;
    vl powB;

    PowerMemo(ll sz, ll base, ll _mod):
        sz(sz), base(base), mod(_mod){
            powB.resize(sz + 5);
            init();
        }

    void init(){
        powB[0] = 1;
        rep(i, sz + 3)
            powB[i + 1] = (powB[i] * base) % mod;
    }

    ll operator[](ll k){
        return powB[k];
    }
};

struct Grid2D{
    Graph graph;
    ll Width, Height;

    Grid2D(ll w, ll h) : Width(w), Height(h){
        graph.resize(w * h);
    }

    ll pos_to_idx(point p){
        return p.y * Width + p.x;
    }

    point idx_to_pos(ll idx){
        return {idx % Width, idx / Width};
    }

    bool undefined_region(point p, vb2& block){
        if (p.x < 0 || p.x > Width - 1) return true;
        if (p.y < 0 || p.y > Height - 1) return true;
        if (block[p.x][p.y]) return true;
        return false;
    }

    void build(vb2& block, ll val = 1){
        rep(x, Width) rep(y, Height){
            point p = {x, y};
            ll idx1 = pos_to_idx(p); ll idx2;

            if (block[x][y]) continue;
            rep(i, 4){
                point nxt = p + dirs[i];
                idx2 = pos_to_idx(nxt);
                if (!undefined_region(nxt, block))
                    graph[idx1].pb({idx2, val}); // dist[idx1][idx2] = val; (warshall-floyd)
            }
        }
    }
};

struct Cumulative2D{
    vl2 cum;
    ll w, h;

    Cumulative2D(ll w, ll h) : w(w), h(h){
        cum = vl2(w + 1, vl(h + 1, 0));
    }

    template<typename T>
    void build(vector<T>& vec){
        // never forget building
        rep(x, w + 1) cum[x][0] = 0;
        rep(y, h + 1) cum[0][y] = 0;
        rep(y, h) rep(x, w)
            cum[x + 1][y + 1] = cum[x][y + 1] + vec[x][y];
        rep(x, w + 1) rep(y, h)
            cum[x][y + 1] += cum[x][y];
    }

    ll func(ll x, ll y, ll dx, ll dy){
        // 1-indexed
        // the rectangle of (x, y), (x + dx, y), (x, y + dy) and (x + dx, y + dy)
        // think about the case of (1, 1, 1, 1).
        if (x + dx > w || y + dy > h) return - INF;
        ll val = cum[x + dx][y + dy];
        val += cum[x][y];
        val -= cum[x][y + dy];
        val -= cum[x + dx][y];
        return val;
    }
};

ll gcd(ll m, ll n){
    ll a = max(m, n);
    ll b = min(m, n);
    while(b != 1 && b != 0){ a %= b; swap(a, b);}
    return b == 1 ? 1 : a;
}

ll lcm(ll m, ll n){ return m / gcd(m, n) * n;}

ll power_normal(ll a, ll power){
    ll value = 1;
    while(power != 0){
        if(power & 1) value = value * a;
        a = a * a;
        power = power >> 1;
    }
    return value;
}

ll power_mod(ll a, ll power, ll mod){
    ll value = 1;
    while(power != 0){
        if(power & 1) value = (value * a) % mod;
        a = (a * a) % mod;
        power = power >> 1;
    }
    return value % mod;
}

ll modinv(ll a, ll mod){
    return power_mod(a, mod - 2, mod);
}

ll combination(ll n, ll r, ll mod){
    if (n == r && n == 0) return 1;
    else if (n <= 0 || r < 0 || r > n) return 0;
    ll numerator = 1;
    ll denomenator = 1;
    for(ll i = 0; i < r ; i++){
        ll num = (n - i) % mod, den = (i + 1) % mod;
        (numerator *= num) %= mod;
        (denomenator *= modinv(den, mod)) %= mod;
    }

    return (numerator * denomenator) % mod;
}

vl2 pascal_triangle(ll n){
    /*
    Complexity: O(n^2)
    The upper bound of n is nearly 50.

    Parameters
    ----------
    n; the size of returned vector

    Returns
    -------
    comb[i][j]: combination(i, j). 0 <= i <= n, 0 <= j <= i
    */

    vl2 comb(n + 1, vl(n + 1));
    comb[0][0] = 1;
    For(i, 1, n + 1) rep(j, i + 1){
        comb[i][j] += comb[i - 1][j];
        if (j > 0) comb[i][j] += comb[i - 1][j - 1];
    }

    return comb;
}

ld log_combination(ll n, ll r){
    if (n == r && n == 0) return 0;
    else if (n <= 0 || r < 0 || r > n) return -INF;
    ld val = 0;
    for(ll i = 0; i < r ; i++){
        val += log(n - i);
        val -= log(i + 1);
    }

    return val;
}

str bin_expression(ll n, ll dig){
    str s = "";
    rep(i, dig){
        s += ltos(n % 2);
        n /= 2;
    }
    reverse(all(s));
    return s;
}

bool is_prime(ll n){
    if (n <= 1) return false;
    for(ll i = 2; i * i <= n; i++){
        if (n % i == 0) return false;
    }
    return true;
}

mll prime_factorization(ll n){
    ll i = 2;
    mll table;
    while(i * i <= n){
        while(n % i == 0){
            table[i] ++;
            n /= i;
        }
        i++;
    }
    if (n > 1) table[n] = 1;
    return table;
}

vl divisor_table(ll n){
    vl table;
    ll i = 1;
    while(i * i <= n){
        if (n % i == 0){
            table.pb(i);
            if (i * i != n) table.pb(n / i);
        }
        i++;
    }

    sort(all(table));

    return table;
}

ll next_combination(ll sub){
    /*
    nCk
    ll bit = (1 << k) - 1;
    for (; bit < (1 << n); bit = next_combination(bit)){
        bool ith = bit & (1 << i);
        procedures...
    }

        sub & -sub: the binary which shares the last digit whose value is 1 in sub
        sub + x   : carry up the last digit
            ~y    : the binary whose digits are 1 if y's digit is 0.
    (sub & ~y) / x: reduce the same number of 0s after first 1 in x from (sub & ~y).
    */
    ll x = sub & -sub, y = sub + x;

    if (x != 0)
        return (((sub & ~y) / x) >> 1) | y;
    else
        return INF;
}

// just change the input if you want to change the target.
// If you want to check the common sequences in two strings,
// combine them. e.g. ABC150F
template<typename T>
vl z_algorithm(T& s, ll n){
    /*
    Paramters
    ---------
    T: the string or list of interest
    n: the size of string or list

    Returns
    -------
    res[i] is the maximum number of K which satisfies
    s[:K] == s[i:i + K]
    for each i = 0, 1, 2, ..., n - 1.
    */
    vl res(n);
    res[0] = n;
    ll i1 = 1, i2 = 0;

    while(i1 < n){
        /*
        i1: the starting point
        i2: the length of substring
        */
        while (i1 + i2 < n && s[i2] == s[i1 + i2]) ++i2;
        res[i1] = i2; if (i2 == 0){++i1; continue;}
        ll i3 = 1;

        // update the already seen points
        while (i1 + i3 < n && i3 + res[i3] < i2){res[i1 + i3] = res[i3]; ++i3;}
        // update up to i1 + i3 and the next possible minimum length is i2 - i3 (= res[i3])
        i1 += i3, i2 -= i3;
    }
    return res;
}

ll big_number_mod(str s, ll mod){
    ll l = s.length(); ll idx = 0;
    ll val = 0; ll tenth = 1;
    while (idx < l){
        ll m = s[l - 1 - idx] - '0';
        val += (m * tenth) % mod; val %= mod;
        tenth *= 10; tenth %= mod; idx++;
    }

    return val;
}

ll big_number_compare(str s1, str s2){
    if (s1.length() > s2.length()) return 1;
    else if (s1.length() < s2.length()) return -1;
    else if (s1 == s2) return 0;

    return 2 * (s1 > s2) - 1;
}

ll string_to_ll(str s){
    ll l = s.length(); ll idx = 0;
    ll val = 0; ll tenth = 1;
    while (idx < l){
        ll m = s[l - 1 - idx] - '0';
        val += (m * tenth);
        tenth *= 10; idx++;
    }

    return val;
}

str reflected_string(str s){
    str t, u; ll n = s.length();
    t = s; reverse(all(t));
    u = substring(t, 0, n - 2) + s + substring(t, 1, n - 1);
    return u;
}

ld distance_between_point_line(point l_begin, point l_end, point p){
    ll xl1 = l_begin.x, yl1 = l_begin.y;
    ll xl2 = l_end.x, yl2 = l_end.y;
    ll xp = p.x, yp = p.y;
    ll a = yl2 - yl1;
    ll b = - xl2 + xl1;
    ll c = - a * xl2 - b * yl2;
    return abs(ld(a * xp + b * yp + c)) / ld(sqrt(a * a + b * b));
}

bool is_cross(point l1_begin, point l1_end, point l2_begin, point l2_end){
    ll x1 = l1_begin.x, y1 = l1_begin.y;
    ll x2 = l1_end.x, y2 = l1_end.y;
    ll x3 = l2_begin.x, y3 = l2_begin.y;
    ll x4 = l2_end.x, y4 = l2_end.y;
    ll val1 = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3);
    ll val2 = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4);
    ll val3 = (x3 - x4) * (y1 - y3) + (y3 - y4) * (x3 - x1);
    ll val4 = (x3 - x4) * (y2 - y3) + (y3 - y4) * (x3 - x2);

    return val1 * val2 < 0 && val3 * val4 < 0;
}

template<typename T>
bool isColinear(T p1, T p2, T p3){
    T v1 = p2 - p1, v2 = p3 - p1;
    return v1.x * v2.y == v1.y * v2.x;
}

template<typename T>
T PerpendicularBisector(T p1, T p2){
    T vec = p2 - p1;
    assert(!vec.is_zero());
    T ret = {vec.y, -vec.x};

    return ret;
}

template<typename T>
ld Distance2DPoints(T p1, T p2){
    T vec = (p1 - p2) * (p1 - p2);
    return sqrt(vec.x + vec.y);
}

ll SquaredDistance2DPoints(point p1, point p2){
    point vec = (p1 - p2) * (p1 - p2);
    return vec.x + vec.y;
}

ld space_of_triangle(point p1, point p2, point p3){
    ll x1 = p1.x, y1 = p1.y;
    ll x2 = p2.x, y2 = p2.y;
    ll x3 = p3.x, y3 = p3.y;
    ll v1 = x2 - x1; ll u1 = y2 - y1;
    ll v2 = x3 - x1; ll u2 = y3 - y1;
    ld s = ld(v1 * u2 - u1 * v2) / ld(2);
    return abs(s);
}

pair<point, ll> OuterCenter(point p1, point p2, point p3){
    // the center of circle on the given three points
    // return the determinant value and the product of center points and 2 * determinant value
    point ret;

    if (isColinear(p1, p2, p3)){
        ll d1 = SquaredDistance2DPoints(p1, p2);
        ll d2 = SquaredDistance2DPoints(p2, p3);
        ll d3 = SquaredDistance2DPoints(p3, p1);
        if (d1 >= d2 && d1 >= d3) {ret = p1 + p2; return mp(ret, 2);}
        else if (d2 >= d1 && d2 >= d3) {ret = p2 + p3; return mp(ret, 2);}
        else {ret = p3 + p1; return mp(ret, 2);}
    }

    point pv1 = PerpendicularBisector(p1, p2);
    point pv2 = PerpendicularBisector(p1, p3);
    point cv1_2x = p1 + p2, cv2_2x = p1 + p3;
    // cv1 + k pv1 == cv2 + m pv2
    // (pv1x -pv2x) (k) = (cv2x - cv1x)
    // (pv1y -pv2y) (m) = (cv2y - cv1y)
    ll det_inv = - pv1.x * pv2.y + pv1.y * pv2.x;
    ll x1_2x = cv2_2x.x - cv1_2x.x, x2_2x = cv2_2x.y - cv1_2x.y;
    pl c_2x_det = {-pv2.y * x1_2x + pv2.x * x2_2x, -pv1.y * x1_2x + pv1.x * x2_2x};
    
    // ret.x = ld(cv1_2x.x * det_inv + pv1.x * c_2x_det.e1) / ld(2 * det_inv);
    // ret.y = ld(cv1_2x.y * det_inv + pv1.y * c_2x_det.e1) / ld(2 * det_inv);
    ret.x = cv1_2x.x * det_inv + pv1.x * c_2x_det.e1;
    ret.y = cv1_2x.y * det_inv + pv1.y * c_2x_det.e1;

    ll jacobian = 2 * det_inv;
    return mp(ret, jacobian);
}

ll inversion_number(vl a, ll a_max){
    /*
    Paramters
    ---------
    a: vector<ll>
        All the elements must be non-negative.
        Prefably the elements are compressed to reduce the computational cost.
    a_max: ll
        The maximum value of the vector a or the value bigger than the value stated previously.
    */

    BinaryIndexedTree bit(a_max + 1); ll val = 0;
    rep(i, a.size()){
        // i is the number of elements that have lower index than a[i].
        // call the number of elements that have lower value than a[i]
        // by subtracting these two, the residual number is the number of elements that have larger value.
        val += i - bit.query(a[i] - 1);// cumulative sum from 0 to a[i] - 1
        bit.update(a[i], 1);
    }
    return val;
}

ld bin_search(ld left, ld right, bool lb, function<bool(ld)> judge){
    ld mid;
    while(right - left > EPS){
        mid = (right + left) / 2;
        if (lb){
            if (judge(mid)) right = mid;
            else left = mid + EPS;
        }else{
            if (judge(mid)) left = mid;
            else right = mid - EPS;
        }
    }
    return right;
}

ll bin_search(ll left, ll right, bool lb, function<bool(ll)> judge){
    ll mid;
    while(right > left){
        if (lb){
            // if true (satisfies the condition), range shifts smaller direction
            mid = (right + left) / 2;
            if (judge(mid)) right = mid;
            else left = mid + 1;
        }else{
            // if true (satisfies the condition), range shitfs larger direction
            mid = (right + left + 1) / 2;
            if (judge(mid)) left = mid;
            else right = mid - 1;
        }
    }
    return right;
}

ld trinary_search(ld left, ld right, function<ld(ld)> func){
    // Care the value EPS!!! Compare to the condition
    while(abs(right - left) > EPS){
        ld left2 = (2.0 * left + right) / 3.0;
        ld right2 = (left + 2.0 * right) / 3.0;
        ld f1 = func(left2);
        ld f2 = func(right2);
        if (f1 <= f2) right = right2;
        else if (f2 <= f1) left = left2;
    }

    return right;
}

template<typename T>
vector<T> compress(vector<T> v){
    // sort and remove all the duplicated values
    sort(all(v));
    v.erase(unique(all(v)), v.end());
    return v;
}

template<typename T>
map<T, ll> dict(const vector<T> &v){
    map<T, ll> d;
    rep(i, v.size()) d[v[i]] = i;
    return d;
}

points compress2D(vl xs, vl ys){
    /*
    NOTE
    ----
    Add the corner points if required
    */

    ll n = xs.size();
    vl xcs = compress(xs), ycs = compress(ys);
    map<ll, ll> xd = dict(xcs), yd = dict(ycs);
    points ps(n);

    rep(i, n)
        xs[i] = xd[xs[i]], ys[i] = yd[ys[i]];

    rep(i, n)
        ps[i] = {xs[i], ys[i]};

    sort(all(ps));

    return ps;
}

void GaussJordanBitVector(vl &bs){
    ll n = bs.size();
    ll rank = 0; ll j = 0;
    revrep(i, N_DIGITS){
        for(j = rank; j < n; j++) if (bs[j] & (1LL << i)) break;
        if (j == n) continue;
        if (j > rank) bs[rank] ^= bs[j];
        for (j = rank + 1; j < n; j++) bs[j] = min(bs[j], bs[j] ^ bs[rank]);
        rank++;
    }
}

ll kruskal(vector<undirected_edge>& es, ll n_vertex){
    // O(ElogE)
    sort(all(es));
    UnionFind uf(n_vertex);

    ll min_cost = 0;

    rep(i, es.size()){
        undirected_edge& e = es[i];
        if (!uf.is_same(e.from, e.to)){
            min_cost += e.cost;
            uf.unite(e.from, e.to);
        }
    }
    return min_cost;
}

ll LongestIncreasedSequence(vl& v){
    ll n = v.size();
    vl dp(n, INF);
    rep(i, n)
        *lower_bound(all(dp), v[i]) = v[i];
    return lower_bound(all(dp), INF) - dp.begin();
}

void dijkstra(ll start, Graph& graph, vl& dist,
              vl& vertex_pre, bool trace = false){
    priority_queue<pl, vpl, greater<pl>> edge_costs;
    ll n = graph.size();
    dist = vl(n, INF);
    if (trace) vertex_pre = vl(n, -1);
    dist[start] = 0;
    edge_costs.push(pl(0, start));

    while (!edge_costs.empty()){
        ll idx, cost;
        tie(cost, idx) = edge_costs.top();
        edge_costs.pop();
        if (dist[idx] < cost) continue;

        for (auto e: graph[idx]){
            if (dist[e.to] > dist[idx] + e.cost){
                dist[e.to] = dist[idx] + e.cost;
                if (trace) vertex_pre[e.to] = idx;
                edge_costs.push(pl(dist[e.to], e.to));
            }
        }
    }
}

vl get_predecessor(ll g, vl& vertex_pre){
    vl path;
    for (; g != -1; g = vertex_pre[g]) path.pb(g);

    reverse(all(path));
    return path;
}

void warshall_floyd(vl2& dist){
    ll n = dist.size();
    // Dont forget the initialization
    // rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(k, n) rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j]);
}

// ABC061D
bool find_negative_cycle(ll n, ll goal, Graph& graph, vl& dist){
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[e.to] != INF && dist[e.to] > dist[v] + e.cost){
            dist[e.to] = -INF;
            if (goal == -1) return true;
            else if (goal == e.to) return true;
        }
    }
    return false;
}

bool bellman_ford(ll start, ll goal, Graph& graph, vl& dist){
    // if there is a closed circuit, it returns false. (when goal == -1)
    // if the distance to goal cannot be obtained, it returns false (when goal != -1)
    ll n = graph.size();
    dist = vl(n, INF);
    dist[start] = 0;
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[v] != INF && dist[e.to] > dist[v] + e.cost)
            dist[e.to] = dist[v] + e.cost;
    }

    if (find_negative_cycle(n, goal, graph, dist)) return false;
    return true;
}

void Euler_Tour(Graph& tree, vl& euler_tour, vl& in, vl& out, ll anc = 0){
    // record only when we first reach a node and leave the node.
    ll n = tree.size();
    vb visited(n, false);
    in = out = vl(n);

    auto dfs = [&](auto dfs, ll source) -> void{
        visited[source] = true;
        in[source] = euler_tour.size();
        euler_tour.pb(source);

        bool is_leaf = true;
        for (auto& e: tree[source]){
            ll target = e.to;
            if (visited[target]) continue;
            else is_leaf = false;
            dfs(dfs, target);
        }
        
        euler_tour.pb(- source);
        out[source] = euler_tour.size() - 1;
    };

    dfs(dfs, anc);
}

void Euler_Tour2(Graph& tree, vl& euler_tour, vl& in, vl& out, ll anc = 0){
    // record everytime we reach a node
    ll n = tree.size();
    vb visited(n, false);
    in = out = vl(n);

    auto dfs = [&](auto dfs, ll source) -> void{
        visited[source] = true;
        in[source] = euler_tour.size();
        euler_tour.pb(source);

        bool is_leaf = true;
        for (auto& e: tree[source]){
            ll target = e.to;
            if (visited[target]) continue;
            else is_leaf = false;
            dfs(dfs, target);
            euler_tour.pb(source);
        }

        out[source] = euler_tour.size() - 1;
    };

    dfs(dfs, anc);
}

std::mt19937 create_rand_engine(){
    std::random_device rnd;
    std::vector<std::uint_least32_t> v(10);
    std::generate(v.begin(), v.end(), std::ref(rnd));
    std::seed_seq seed(v.begin(), v.end());
    return std::mt19937(seed);
}


vl generate_unique_array(const size_t sz, ll vm, ll vM){
    const size_t range = static_cast<size_t>(vM - vm + 1);
    const size_t num = static_cast<size_t>(sz * 1.2);
    assert(vm <= vM);
    assert(sz <= range);

    vl ret;
    auto engine = create_rand_engine();
    std::uniform_int_distribution<ll> distribution(vm, vM);

    while (ret.size() < sz){
        while (ret.size() < num) ret.pb(distribution(engine));
        sort(all(ret));

        auto unique_end = unique(all(ret));
        if (sz < distance(ret.begin(), unique_end)) 
            unique_end = next(ret.begin(), sz);

        ret.erase(unique_end, ret.end());
    }

    return ret;
}

vl generate_array(const size_t sz, ll vm, ll vM){
    const size_t range = static_cast<size_t>(vM - vm + 1);
    assert(vm <= vM);

    vl ret;
    auto engine = create_rand_engine();
    std::uniform_int_distribution<ll> distribution(vm, vM);

    while (ret.size() < sz)
        ret.pb(distribution(engine));

    return ret;
}

void read_graph(ll n_vertex, ll n_edges, Graph& graph,
                ll cost = -1, bool directed = false, ll offset = -1){

    graph.resize(n_vertex);
    rep(i, n_edges){
        ll from, to, c = cost;
        if (cost == -1) in3(from, to, c);
        else in2(from, to);
        from += offset, to += offset;

        graph[from].pb({to, cost});
        if (!directed) graph[to].pb({from, cost});
    }
}

void read_vector(ll n, vl& v, ll offset = 0){
    v.resize(n);
    rep(i, n){
        in1(v[i]);
        v[i] += offset;
    }
}

void read_pair_vector(ll n, vpl& v, ll offset1 = 0, ll offset2 = 0, bool rev = false){
    v.resize(n);
    rep(i, n){
        ll a, b; in2(a, b);
        a += offset1, b += offset2;
        if (!rev) v[i] = {a, b};
        else v[i] = {b, a};
    }
}

void read_2DMap(ll w, ll h, vb2& block, char b){
    block = vb2(w, vb(h, false));
    str s;
    rep(y, h){
        in1(s);
        rep(x, w) block[x][y] = (s[x] == b);
    }
}

/*

diameter of tree
Graph tree;
ll dM = 0, vM = 0, v2 = 0;

void dfs1(ll source, ll parent, ll d){
    if (d > dM) dM = d, vM = source;

    rep(i, tree[source].size()){
        ll target = tree[source][i].to;
        if (target == parent) continue;
        dfs1(target, source, d + 1);
    }
}

void dfs2(ll source, ll parent, ll d){
    if (dM <= d) dM = d, v2 = source;
    rep(i, tree[source].size()){
        ll target = tree[source][i].to;
        if (target == parent) continue;
        dfs2(target, source, d + 1);
    }
}

dfs(0, -1, 0);
dfs2(vM, -1, 0);
prl2(vM + 1, v2 + 1); // the two edges of tree

*/

/*

    # 2. The usage of next_permutation and combination (factorial search)
    ll a[8];
    rep(i, 8) a[i] = i;
    sort(a, a + 8);
    do{
    }while(next_permutation(a, a+n));

    # 4. imos method
    // used when we would like to count the number which
    // shows how many times the numbers between l and r belongs to smt.
    // This method is composed of three process.
    ll n, m, s[MAX_M], l, r;
    in2(n, m);
    rep(i, m) s[i] = 0;

    // 1st step
    rep(i, n){
        in3(l, r, c);
        l--; r--; // if l starts from 1.
        s[l] += c; s[r + 1] -= c;
    }

    // 2nd step
    rep(i, m - 1) s[i + 1] += s[i];
    // 3rd step: judgement...

    #5. shakutori method (syakutori, two pointers technique)
    // 1. strech right side while the condition is met.
    // 2. renew the answer
    // 3. increments left side
    // 4. Back to 1. (l <= r must be satisfied all the time.)

    ll l = 0; ll r = 0;
    while (l < n){
        if (l == r) r++;
        while(r < n && cond) r++;
        l++;
    }
    prl(answer);

    #11. bfs ABC146D, ABC007C
    1. first create a tree.
    2. start searching from a node.
    3. do some processes and push nodes connected with a given target node in BFS.
    4. repeat a series of procedure until queue is empty.

    queue<pl> q;

    void bfs(ll source, ll parents){
        ll n_edge = graph[source].size();
        if (parents != -1) dist[source] = min(dist[source], dist[parents] + 1);
        if (visited[source]) return;
        visited[source] = true;

        rep(idx, n_edge){
            ll target = graph[source][idx].to;
            if (target == parents) continue;
            q.push(mp(target, source));
        }
    }

    q.push(mp(sg.e1, -1));
    while(!q.empty()){
        pl source = q.front(); q.pop();
        bfs(source.e1, source.e2);
    }
*/

/*
     x1: 0000 0001 0010 0101 0110 0111 0111
     x2: xxxx 0001 0011 0100 0101 1000 0110
x1 & x2: 0000 0001 0010 0100 0100 0000 0110

      x: 1001 1010  1100 1011 1101  1111
x & ~ x: 0001 0010  0100 0001 0001  0001
    sum: 1010 1100 10000 1100 1100 10000

####### Attention #######
S & (1 << i) -> if true, i in S
S | (1 << i) -> S union {i}
S & ~(1 << i) -> S - {i}
__builtin_popcountl(i) -> the number of 1 in binary

#Conditional Operator
condition ? true : false;

# inclusion-exclusion principle
|U[i = 1 to n] Ai| = sum[i = 1 to n] |Ai| - sum[i < j]|Ai ^ Aj| + ... + (-1)^(n - 1) |^[i = 1 to n]Ai|
*/

const ll MAX_N = 200005;
const size_t MAX_BIT = 100;
typedef bitset<MAX_BIT> bts;
typedef vector<bts> vbt;
typedef vector<vbt> vbt2;
typedef vector<vbt2> vbt3;
bool okay = false;
ll answer = 0;
str s, t, u;
ll n, m, h, w, k, Q;
ll a, b, c, d, idx, cnt;
vl as, bs, cs, ds;
vb2 block;
vb visited;
Graph graph;
Graph tree;
/* 
read_graph(n, m, graph, cost, directed, offset)
    cost: if link cost is fixed, put positive integer
          otherwise -1
    directed: if directed true, olse false
    offset: if the given vertex is 1-indexed, -1.
*/

void solve(){
    // generate_array(size, min, max);
    in1(n);
    multiset<pl> rs, ls;
    rep(i, n){
        in1(s);
        ll cur = 0, bottom = 0;
        rep(j, s.length()){
            cur += (s[j] == '(') * 2 - 1;
            chmin(bottom, cur);
        }
        if (cur >= 0) ls.insert({bottom, cur});
        else rs.insert({bottom - cur, -cur});
    }

    ll lv = 0, rv = 0;
    auto it = ls.rbegin();
    auto update = [&](ll& val) -> bool{
        pl p = *(it++);
        if (p.e1 + val >= 0) {val += p.e2; return true;}
        else return false;
    };

    while(it != ls.rend()) if (!update(lv)) {prl("No"); return;}
    it = rs.rbegin();
    while(it != rs.rend()) if (!update(rv)) {prl("No"); return;}
    yn(lv == rv);

    // ### DEBUG PART ###
    auto naive_solver = [&](){
    };
    #ifdef _LOCAL
        naive_solver();
    #endif
}

void test(ll num = 0, bool verbose = false){
    rep(i, max(1LL, num)){
        ll t = clock();
        if (verbose) prl4("\n#####", i + 1, "#####", "\n## Answer ##");
        solve();
        if (verbose) {prl(""); prl_time(t);}
    }
}

int main(void){
    #ifdef _LOCAL
        test(4, true);
    #else
        test(0, false);
    #endif

    return 0;
}


Measured memory_bytes: 66204