Language: C++

//#pragma GCC optimize("Ofast")
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace std::chrono;
#define int long long
#define ll long long
auto start_time = system_clock::now();

#ifdef _DEBUG
//区間削除は出来ない
template<class T> struct my_pbds_tree {
    set<T> s;
    auto begin() { return s.begin(); }
    auto end() { return s.end(); }
    auto rbegin() { return s.rbegin(); }
    auto rend() { return s.rend(); }
    auto empty() { return s.empty(); }
    auto size() { return s.size(); }
    void clear() { s.clear(); }
    template<class U> void insert(U v) { s.insert(v); }
    template<class U> void operator+=(U v) { insert(v); }
    template<class F> auto erase(F v) { return s.erase(v); }
    template<class U> auto find(U v) { return s.find(v); }
    template<class U> auto lower_bound(U v) { return s.lower_bound(v); }
    template<class U> auto upper_bound(U v) { return s.upper_bound(v); }
    auto find_by_order(long long k) {
        auto it = s.begin();
        for (long long i = 0; i < k; i++)it++;
        return it;
    }
    auto order_of_key(long long v) {
        auto it = s.begin();
        long long i = 0;
        for (; it != s.end() && *it < v; i++)it++;
        return i;
    }
};
#define pbds(T) my_pbds_tree<T>
//gp_hash_tableでcountを使えないようにするため
template<class T, class U> struct my_unordered_map {

    unordered_map<T, U> m;
    my_unordered_map() {};
    auto begin() { return m.begin(); }
    auto end() { return m.end(); }
    auto cbegin() { return m.cbegin(); }
    auto cend() { return m.cend(); }
    template<class V> auto erase(V v) { return m.erase(v); }
    void clear() { m.clear(); }    /*countは gp_hash_tableに存在しない*/    /*!= m.end()*/    template<class V> auto find(V v) { return m.find(v); }
    template<class V> auto &operator[](V n) { return m[n]; }
};
#define unordered_map my_unordered_map
#define umapi unordered_map<long long,long long>
#define umapp unordered_map<P,long long>
#define umapip unordered_map<long long,P>

#else
#define unordered_map __gnu_pbds::gp_hash_table
//find_by_order(k) k番目のイテレーター
//order_of_key(k)  k以上が前から何番目か
#define pbds(U) __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>

#define umapi unordered_map<long long,long long,xorshift>
#define umapp unordered_map<P,long long,xorshift>
#define umapip unordered_map<long long,P,xorshift>

#endif
struct xorshift {
    static uint64_t splitmix64(uint64_t x) {
        x += 0x9e3779b97f4a7c15;
        x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
        x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
        return x ^ (x >> 31);
    }
    size_t operator()(uint64_t x) const {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }
    size_t operator()(std::pair<long long, long long> x) const {
        long long v = ((x.first) << 32) | x.second;
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(v + FIXED_RANDOM);
    }
};
template<class U, class L> void operator+=(__gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update> &s, L v) { s.insert(v); }
//衝突対策
#define ws wszzzz

template<class A, class B, class C> struct T2 { A f;B s;C t;T2() { f = 0, s = 0, t = 0; }T2(A f, B s, C t) : f(f), s(s), t(t) {}bool operator<(const T2 &r) const { return f != r.f ? f < r.f : s != r.s ? s < r.s : t < r.t;        /*return f != r.f ? f > r.f : s != r.s ?n s > r.s : t > r.t; 大きい順 */   } bool operator>(const T2 &r) const { return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t;        /*return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 小さい順 */   } bool operator==(const T2 &r) const { return f == r.f && s == r.s && t == r.t; } bool operator!=(const T2 &r) const { return f != r.f || s != r.s || t != r.t; }};
template<class A, class B, class C, class D> struct F2 {
    A a;
    B b;
    C c;
    D d;
    F2() { a = 0, b = 0, c = 0, d = 0; }
    F2(A a, B b, C c, D d) : a(a), b(b), c(c), d(d) {}
    bool operator<(const F2 &r) const { return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c < r.c : d < r.d;    /*    return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c > r.c : d > r.d;*/    }
    bool operator>(const F2 &r) const { return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c > r.c : d > r.d;/*        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c < r.c : d < r.d;*/    }
    bool operator==(const F2 &r) const { return a == r.a && b == r.b && c == r.c && d == r.d; }
    bool operator!=(const F2 &r) const { return a != r.a || b != r.b || c != r.c || d != r.d; }
    long long operator[](long long i) {
        assert(i < 4);
        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;
    }
};
typedef T2<long long, long long, long long> T;
typedef F2<long long, long long, long long, long long> F;
T mt(long long a, long long b, long long c) { return T(a, b, c); }

//@マクロ省略系 型,構造
#define double long double
#define ull unsigned long long
using dou = double;
using itn = int;
using str = string;
using bo= bool;
#define au auto
using P = pair<long long, long long>;
#define fi first
#define se second
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==
#define el else
#define elf else if
#define wh while
#define upd update

#define maxq 1
#define minq -1

#define ZERO(a) memset(a,0,sizeof(a))
#define MINUS(a) memset(a,0xff,sizeof(a))
#define MALLOC(type, len) (type*)malloc((len) * sizeof(type))
#define lam(right) [&](long long& p){return p right;}

//マクロ省略系 コンテナ
using vi = vector<int>;
using vb = vector<bool>;
using vs = vector<string>;
using vd = vector<double>;
using vc = vector<char>;
using vp = vector<P>;
using vt = vector<T>;

#define vec vector
#define o_vvt(o1, o2, o3, o4, name, ...) name
#define vvt0(t) vec<vec<t>>
#define vvt1(t, a) vec<vec<t>>a
#define vvt2(t, a, b) vec<vec<t>>a(b)
#define vvt3(t, a, b, c) vec<vec<t>> a(b,vec<t>(c))
#define vvt4(t, a, b, c, d) vec<vec<t>> a(b,vec<t>(c,d))

#define vvi(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(long long,__VA_ARGS__)
#define vvb(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(bool,__VA_ARGS__)
#define vvs(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(string,__VA_ARGS__)
#define vvd(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(double,__VA_ARGS__)
#define vvc(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(char,__VA_ARGS__)
#define vvp(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(P,__VA_ARGS__)
#define vvt(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(T,__VA_ARGS__)

template<typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template<typename T, typename... Ts> auto make_v(size_t a, Ts... ts) { return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...)); }
#define vni(name, ...) auto name = make_v<long long>(__VA_ARGS__)
#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)
#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)
#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)
#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)
#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)

#define PQ priority_queue<long long, vector<long long>, greater<long long> >
#define tos to_string
using mapi = map<long long, long long>;
using mapp = map<P, long long>;
using mapd = map<dou, long long>;
using mapc = map<char, long long>;
using maps = map<str, long long>;
using seti = set<long long>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<long long>;
#define bset bitset
#define uset unordered_set
#define useti unordered_set<long long,long long,xorshift>
#define mset multiset
#define mseti multiset<long long>
#define umap unordered_map
#define mmap multimap

template<class T> struct pq {
    priority_queue<T, vector<T>, greater<T> > q;/*小さい順*/    T su = 0;
    void clear() {
        q = priority_queue<T, vector<T>, greater<T> >();
        su = 0;
    }
    void operator+=(T v) {
        su += v;
        q.push(v);
    }
    T sum() { return su; }
    T top() { return q.top(); }
    void pop() {
        su -= q.top();
        q.pop();
    }
    T poll() {
        T ret = q.top();
        su -= ret;
        q.pop();
        return ret;
    }
    int size() { return q.size(); }
};
template<class T> struct pqg {
    priority_queue<T> q;/*大きい順*/    T su = 0;
    void clear() {
        q = priority_queue<T>();
        su = 0;
    }
    void operator+=(T v) {
        su += v;
        q.push(v);
    }
    T sum() { return su; }
    T top() { return q.top(); }
    void pop() {
        su -= q.top();
        q.pop();
    }
    T poll() {
        T ret = q.top();
        su -= ret;
        q.pop();
        return ret;
    }
    int size() { return q.size(); }
};
#define pqi pq<int>
#define pqgi pqg<int>
//マクロ 繰り返し
#define o_rep(o1, o2, o3, o4, name, ...) name
# define rep1(n) for(int rep1i = 0,rep1lim=n; rep1i < rep1lim ; ++rep1i)
# define rep2(i, n) for(int i = 0,rep2lim=n; i < rep2lim ; ++i)
#define rep3(i, m, n) for(int i = m,rep3lim=n; i < rep3lim ; ++i)
#define rep4(i, m, n, ad) for(int i = m,rep4lim=n; i < rep4lim ; i+= ad)
#define rep(...) o_rep(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)

#define rer2(i, n) for(int i = n; i >= 0 ; i--)
#define rer3(i, m, n) for(int i = m,rer3lim=n; i >= rer3lim ; i--)
#define rer4(i, m, n, dec) for(int i = m,rer4lim=n; i >= rer4lim ; i-=dec)
#define rer(...) o_rep(__VA_ARGS__,rer4,rer3,rer2,)(__VA_ARGS__)

#define reps2(i, j, n) for(int i = 0,reps2lim=n; i < reps2lim ;++i)for(int j = 0; j < reps2lim ; ++j)
#define reps3(i, j, k, n) for(int i = 0,reps3lim=n; i < reps3lim ; ++i)for(int j = 0; j < reps3lim ; ++j)for(int k = 0; k < reps3lim ; ++k)
#define reps4(i, j, k, l, n) for(int i = 0,reps4lim=n; i < reps4lim ; ++i)for(int j = 0; j < reps4lim ; ++j)for(int k = 0; k < reps4lim ; ++k)for(int l = 0; l < reps4lim ; ++l)
#define o_reps(o1, o2, o3, o4, o5, name, ...) name
#define reps(...) o_reps(__VA_ARGS__,reps4,reps3,reps2,rep2,)(__VA_ARGS__)

#define repss(i, j, k, a, b, c) for(int i = 0; i < a ; ++i)for(int j = 0; j < b ; ++j)for(int k = 0; k < c ; ++k)

#define fora(a, b) for(auto&& a : b)
//インデックスを前後含めて走査
#define fori(i, s, len) for (int i = s, prev = (s == 0) ? len - 1 : s - 1, next = (s == len - 1) ? 0 : s + 1, cou = 0; cou < len; cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
//vectorの中身を先頭から見る
#define foriv(i, v, d) int i=0;for (auto prev = d[sz(d)-1],next=d[1],v = d[0]; i < sz(d); i++, prev = v, v = next, next = (i>=sz(d)-1?d[0]:d[i+1]))
#define form(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it).fi < r; ++it)
#define forit(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r;)

//マクロ 定数
#define k3 1010
#define k4 10101
#define k5 101010
#define k6 1010101
#define k7 10101010
const int inf = (int) 1e9 + 100;
const long long linf = (long long) 1e18 + 100;
const char infc = '{';
const string infs = "{";
const double eps = 1e-9;
const double PI = 3.1415926535897932384626433832795029L;

//マクロ省略形 関数等
#define arsz(a) (sizeof(a)/sizeof(a[0]))
#define sz(a) ((int)(a).size())
#define mp make_pair
#define pb pop_back
#define pf push_front
#define eb emplace_back
#define all(a) (a).begin(),(a).end()
#define rall(a) (a).rbegin(),(a).rend()

constexpr bool ev(int a) { return !(a & 1); }
constexpr bool od(int a) { return (a & 1); }
//@拡張系 こう出来るべきというもの

//埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
    template<> class hash<std::pair<signed, signed>> { public:size_t operator()(const std::pair<signed, signed> &x) const { return hash<long long>()(((long long) x.first << 32) | x.second); }};
    template<> class hash<std::pair<long long, long long>> { public:/*大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断*/size_t operator()(const std::pair<long long, long long> &x) const { return hash<long long>()(((long long) x.first << 32) | x.second); }};
}
//stream まとめ
istream &operator>>(istream &iss, P &a) {
    iss >> a.first >> a.second;
    return iss;
}
template<typename T> istream &operator>>(istream &iss, vector<T> &vec) {
    for (T &x: vec) iss >> x;
    return iss;
}
template<class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
    os << p.fi << " " << p.se;
    return os;
}
ostream &operator<<(ostream &os, T p) {
    os << p.f << " " << p.s << " " << p.t;
    return os;
}
ostream &operator<<(ostream &os, F p) {
    os << p.a << " " << p.b << " " << p.c << " " << p.d;
    return os;
}
template<typename T> ostream &operator<<(ostream &os, vector <T> &vec) {
    for (int i = 0; i < vec.size(); ++i)os << vec[i] << (i + 1 == vec.size() ? "" : " ");
    return os;
}
template<typename T, typename U> ostream &operator<<(ostream &os, vector <pair<T, U>> &vec) {
    for (int i = 0; i < vec.size(); ++i) {
        os << vec[i];
        if (i != vec.size() - 1)os << endl;
    }
    return os;
}
template<typename T> ostream &operator<<(ostream &os, vector <vector<T>> &vec) {
    for (int i = 0; i < vec.size(); ++i) {
        for (int j = 0; j < vec[i].size(); ++j) { os << vec[i][j] << " "; }
        os << endl;
    }
    return os;
}
template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {
    for (auto &&v:m) os << v;
    return os;
}
template<class T> ostream &operator<<(ostream &os, set<T> s) {
    fora(v, s) { os << v << " "; }
    return os;
}
template<class T> ostream &operator<<(ostream &os, deque<T> a) {
    fora(v, a)os << v << " ";
    return os;
}
ostream &operator<<(ostream &os, vector <vector<char>> &vec) {
    for(int h = 0,lim=vec.size();h <lim ;h ++){
        for(int  w= 0,lim=vec[0].size(); w< lim; w++){ os << vec[h][w]; }
        os << endl;
    }
    return os;
}
template<typename W, typename H> void resize(vector<W> &vec, const H head) { vec.resize(head); }
template<typename W, typename H, typename ... T> void resize(vector<W> &vec, const H &head, const T ... tail) {
    vec.resize(head);
    for (auto &v: vec)resize(v, tail...);
}
template<typename T, typename F> bool all_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool all_of2(vector<T> &v, F f) {
    for(int i = 0,lim=v.size();i < lim; i++){ if (!all_of2(v[i], f))return false; }
    return true;
}
template<typename T, typename F> bool any_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool any_of2(vector<T> &v, F f) {
    for(int i = 0,lim=v.size();i < lim; i++) { if (any_of2(v[i], f))return true; }
    return false;
}
template<typename T, typename F> bool none_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool none_of2(vector<T> &v, F f) {
    for(int i=0,lim=v.size();i<lim;i++) { if (none_of2(v[i], f))return false; }
    return true;
}
template<typename T, typename F> bool find_if2(T &v, F f) { return f(v); }
template<typename T, typename F> int find_if2(vector<T> &v, F f) {
    for(int i=0,lim=v.size();i<lim;i++) { if (find_if2(v[i], f))return i; }
    return sz(v);
}
template<typename T, typename F> bool rfind_if2(T &v, F f) { return f(v); }
template<typename T, typename F> int rfind_if2(vector<T> &v, F f) {
    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }
    return -1;
}
template<class T> bool contains(string &s, const T &v) { return s.find(v) != string::npos; }
template<typename T> bool contains(vector<T> &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); }
template<typename T, typename F> bool contains_if2(vector<T> &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); }
template<typename T, typename F> int count_if2(T &v, F f) { return f(v); }
template<typename T, typename F> int count_if2(vector<T> &vec, F f) {
    int ret = 0;
    fora(v, vec)ret += count_if2(v, f);
    return ret;
}
template<typename T, typename F> void for_each2(T &v, F f) { f(v); }
template<typename T, typename F> void for_each2(vector<T> &vec, F f) { fora(v, vec)for_each2(v, f); }
template<typename W> int count_od(vector<W> &a) { return count_if2(a, [](int v) { return v & 1; }); }
template<typename W> int count_ev(vector<W> &a) { return count_if2(a, [](int v) { return !(v & 1); }); }
#define all_of(a, right) all_of2(a,lam(right))
#define any_of(a, right) any_of2(a,lam(right))
#define none_of(a, right) none_of2(a,lam(right))
#define find_if(a, right) find_if2(a,lam(right))
#define rfind_if(a, right) rfind_if2(a,lam(right))
#define contains_if(a, right) contains_if2(a,lam(right))
#define count_if(a, right) count_if2(a,lam(right))
#define for_each(a, right) do{fora(v,a){v right;}}while(0)


template<class T, class U> void replace(vector<T> &a, T key, U v) { replace(a.begin(), a.end(), key, v); }
void replace(str &a, char key, str v) { if (v == "")a.erase(remove(all(a), key), a.end()); }
void replace(str &a, char key, char v) { replace(all(a), key, v); }
//keyと同じかどうか01で置き換える
template<class T, class U> void replace(vector<T> &a, U k) { for(int i=0,lim=a.size();i<lim;i++) a[i] = a[i] == k; }
template<class T, class U> void replace(vector<vector<T >> &a, U k) { for(int i=0,lim=a.size();i<lim;i++)for(int j=0,lim=a[0].size();j<lim;j++) a[i][j] = a[i][j] == k; }
template<class T> void replace(T &a) { replace(a, '#'); }
void replace(str &a, str key, str v) {
    stringstream t;
    int kn = sz(key);
    std::string::size_type Pos(a.find(key));
    int l = 0;
    while (Pos != std::string::npos) {
        t << a.substr(l, Pos - l);
        t << v;
        l = Pos + kn;
        Pos = a.find(key, Pos + kn);
    }
    t << a.substr(l, sz(a) - l);
    a = t.str();
}
template<class T> bool includes(vector<T> &a, vector<T> &b) {
    vi c = a;
    vi d = b;
    sort(all(c));
    sort(all(d));
    return includes(all(c), all(d));
}
template<class T> bool is_permutation(vector<T> &a, vector<T> &b) { return is_permutation(all(a), all(b)); }
template<class T> bool next_permutation(vector<T> &a) { return next_permutation(all(a)); }
void iota(vector<int> &ve, int s, int n) {
    ve.resize(n);
    iota(all(ve), s);
}
vi iota(int s, int len) {
    vi ve(len);
    iota(all(ve), s);
    return ve;
}
template<class A, class B> auto vtop(vector<A> &a, vector<B> &b) {
    assert(sz(a) == sz(b));    /*stringを0で初期化できない  */  vector<pair<A, B>> res;
    for(int i=0,lim=a.size();i<lim;i++)res.eb(a[i], b[i]);
    return res;
}
template<class A, class B> void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {
    a.resize(sz(p)), b.resize(sz(p));
    for(int i=0,lim=p.size();i<lim;i++)a[i] = p[i].fi, b[i] = p[i].se;
}
template<class A, class B, class C> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {
    assert(sz(a) == sz(b) && sz(b) == sz(c));
    vector<T2<A, B, C>> res;
    for(int i=0,lim=a.size();i<lim;i++)res.eb(a[i], b[i], c[i]);
    return res;
}
template<class A, class B, class C, class D> auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
    assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));
    vector<F2<A, B, C, D>> res;
    for(int i=0,lim=a.size();i<lim;i++)res.eb(a[i], b[i], c[i], d[i]);
    return res;
}
enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
enum tcomparator {
    fisiti,
    fisitd,
    fisdti,
    fisdtd,
    fdsiti,
    fdsitd,
    fdsdti,
    fdsdtd,
    fitisi,
    fitisd,
    fitdsi,
    fitdsd,
    fdtisi,
    fdtisd,
    fdtdsi,
    fdtdsd,
    sifiti,
    sifitd,
    sifdti,
    sifdtd,
    sdfiti,
    sdfitd,
    sdfdti,
    sdfdtd,
    sitifi,
    sitifd,
    sitdfi,
    sitdfd,
    sdtifi,
    sdtifd,
    sdtdfi,
    sdfdfd,
    tifisi,
    tifisd,
    tifdsi,
    tifdsd,
    tdfisi,
    tdfisd,
    tdfdsi,
    tdfdsd,
    tisifi,
    tisifd,
    tisdfi,
    tisdfd,
    tdsifi,
    tdsifd,
    tdsdfi,
    tdsdfd
};
template<class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {
    typedef pair<A, B> U;
    if (type == fisi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; }); else if (type == fisd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; }); else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; }); else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; }); else if (type == sifi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; }); else if (type == sifd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; }); else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; }); else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
};
template<class U> void sort(vector<U> &a, pcomparator type) { if (type == fisi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; }); else if (type == fisd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; }); else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; }); else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; }); else if (type == sifi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; }); else if (type == sifd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; }); else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; }); else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; }); };
template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D> > &a, pcomparator type) {
    typedef F2<A, B, C, D> U;
    if (type == fisi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; }); else if (type == fisd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; }); else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; }); else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; }); else if (type == sifi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; }); else if (type == sifd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; }); else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; }); else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });
};
template<class U> void sort(vector<U> &a, tcomparator type) {
    if (type == 0) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });
    else if (type == 1) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });
    else if (type == 2) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });
    else if (type == 3) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });
    else if (type == 4) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });
    else if (type == 5) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });
    else if (type == 6) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });
    else if (type == 7) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });
    else if (type == 8) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });
    else if (type == 9) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });
    else if (type == 10) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });
    else if (type == 11) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });
    else if (type == 12) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });
    else if (type == 13) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });
    else if (type == 14) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });
    else if (type == 15) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });
    else if (type == 16) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });
    else if (type == 17) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });
    else if (type == 18) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });
    else if (type == 19) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });
    else if (type == 20) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });
    else if (type == 21) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });
    else if (type == 22) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });
    else if (type == 23) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });
    else if (type == 24) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });
    else if (type == 25) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });
    else if (type == 26) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });
    else if (type == 27) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });
    else if (type == 28) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });
    else if (type == 29) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });
    else if (type == 30) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });
    else if (type == 31) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });
    else if (type == 32) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });
    else if (type == 33) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });
    else if (type == 34) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });
    else if (type == 35) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });
    else if (type == 36) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });
    else if (type == 37) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });
    else if (type == 38) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });
    else if (type == 39) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });
    else if (type == 40) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f; }); else if (type == 41) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f; }); else if (type == 42) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f; }); else if (type == 43) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f; }); else if (type == 44) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f; }); else if (type == 45) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f; }); else if (type == 46) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f; }); else if (type == 47) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });
}
template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {
    typedef F2<A, B, C, D> U;
    if (type == 0) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });
    else if (type == 1) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });
    else if (type == 2) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });
    else if (type == 3) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });
    else if (type == 4) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });
    else if (type == 5) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });
    else if (type == 6) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });
    else if (type == 7) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });
    else if (type == 8) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });
    else if (type == 9) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });
    else if (type == 10) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });
    else if (type == 11) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });
    else if (type == 12) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });
    else if (type == 13) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });
    else if (type == 14) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });
    else if (type == 15) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });
    else if (type == 16) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });
    else if (type == 17) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });
    else if (type == 18) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });
    else if (type == 19) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });
    else if (type == 20) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });
    else if (type == 21) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });
    else if (type == 22) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });
    else if (type == 23) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });
    else if (type == 24) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });
    else if (type == 25) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });
    else if (type == 26) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });
    else if (type == 27) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });
    else if (type == 28) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });
    else if (type == 29) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });
    else if (type == 30) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });
    else if (type == 31) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });
    else if (type == 32) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });
    else if (type == 33) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });
    else if (type == 34) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });
    else if (type == 35) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });
    else if (type == 36) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });
    else if (type == 37) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });
    else if (type == 38) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });
    else if (type == 39) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });
    else if (type == 40) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a; }); else if (type == 41) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a; }); else if (type == 42) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a; }); else if (type == 43) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a; }); else if (type == 44) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a; }); else if (type == 45) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a; }); else if (type == 46) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a; }); else if (type == 47) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });
}

void sort(string &a) { sort(all(a)); }
template<class T> void sort(vector<T> &a) { sort(all(a)); }
//P l, P rで f(P) の形で渡す
template<class U, class F> void sort(vector<U> &a, F f) { sort(all(a), [&](U l, U r) { return f(l) < f(r); }); };
template<class T> void rsort(vector<T> &a) { sort(all(a), greater<T>()); };
template<class U, class F> void rsort(vector<U> &a, F f) { sort(all(a), [&](U l, U r) { return f(l) > f(r); }); };
//F = T<T>
//例えばreturn p.fi + p.se;
template<class A, class B> void sortp(vector<A> &a, vector<B> &b) {
    auto c = vtop(a, b);
    sort(c);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = c[i].fi, b[i] = c[i].se;
}
template<class A, class B, class F> void sortp(vector<A> &a, vector<B> &b, F f) {
    auto c = vtop(a, b);
    sort(c, f); 
    for(int i=0,lim=a.size();i<lim;i++)a[i] = c[i].fi, b[i] = c[i].se;
}
template<class A, class B> void rsortp(vector<A> &a, vector<B> &b) {
    auto c = vtop(a, b);
    rsort(c);
    for(int i=0,lim=a.size();i<lim;i++)a[i] = c[i].first, b[i] = c[i].second;
}
template<class A, class B, class F> void rsortp(vector<A> &a, vector<B> &b, F f) {
    auto c = vtop(a, b);
    rsort(c, f);
    for(int i=0,lim=a.size();i<lim;i++)a[i] = c[i].first, b[i] = c[i].second;
}
template<class A, class B, class C> void sortt(vector<A> &a, vector<B> &b, vector<C> &c) {
    auto d = vtot(a, b, c);
    sort(d);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template<class A, class B, class C, class F> void sortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
    auto d = vtot(a, b, c);
    sort(d, f);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template<class A, class B, class C> void rsortt(vector<A> &a, vector<B> &b, vector<C> &c) {
    auto d = vtot(a, b, c);
    rsort(d);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template<class A, class B, class C, class F> void rsortt(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
    auto d = vtot(a, b, c);
    rsort(d, f);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;
}
template<class A, class B, class C, class D> void sortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
    auto e = vtof(a, b, c, d);
    sort(e);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = e[i].a, b[i] = e[i].b, c[i] = e[i].c, d[i] = e[i].d;
}
template<class A, class B, class C, class D> void rsortf(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {
    auto e = vtof(a, b, c, d);
    rsort(e);
    for(int i=0,lim=a.size();i<lim;i++) a[i] = e[i].a, b[i] = e[i].b, c[i] = e[i].c, d[i] = e[i].d;
}
//sortindex 元のvectorはソートしない
template<class T> vi sorti(vector<T> &a) {
    auto b = a;
    vi ind = iota(0, sz(a));
    sortp(b, ind);
    return ind;
}/*indexの分で型が変わるためpcomparatorが必要*/template<class T> vi sorti(vector<T> &a, pcomparator f) {
    auto b = a;
    vi ind = iota(0, sz(a));
    sortp(b, ind, f);
    return ind;
}
template<class T, class F> vi sorti(vector<T> &a, F f) {
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { return f(a[x]) < f(a[y]); });
    return ind;
}
template<class T> vi rsorti(vector<T> &a) {
    auto b = a;
    vi ind = iota(0, sz(a));
    rsortp(b, ind);
    return ind;
}
template<class T, class F> vi rsorti(vector<T> &a, F f) {
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { return f(a[x]) > f(a[y]); });
    return ind;
}
template<class A, class B, class F> vi sortpi(vector<A> &a, vector<B> &b, F f) {
    auto c = vtop(a, b);
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { return f(c[x]) < f(c[y]); });
    return ind;
}
template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b, pcomparator f) {
    vi ind = iota(0, sz(a));
    auto c = a;
    auto d = b;
    sortt(c, d, ind, f);
    return ind;
}
template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fisi); };
template<class A, class B, class F> vi rsortpi(vector<A> &a, vector<B> &b, F f) {
    auto c = vtop(a, b);
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { return f(c[x]) > f(c[y]); });
    return ind;
}
template<class A, class B> vi rsortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fdsd); };
template<class A, class B, class C, class F> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
    auto d = vtot(a, b, c);
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { return f(d[x]) < f(d[y]); });
    return ind;
}
template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, pcomparator f) {
    vi ind = iota(0, sz(a));
    auto d = vtof(a, b, c, ind);
    sort(d, f);
    for(int i=0,lim=a.size();i<lim;i++)ind[i] = d[i].d;
    return ind;
}
template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c) {
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { if (a[x] == a[y]) { if (b[x] == b[y])return c[x] < c[y]; else return b[x] < b[y]; } else { return a[x] < a[y]; }});
    return ind;
}
template<class A, class B, class C, class F> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {
    auto d = vtot(a, b, c);
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { return f(d[x]) > f(d[y]); });
    return ind;
}
template<class A, class B, class C> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c) {
    vi ind = iota(0, sz(a));
    sort(all(ind), [&](int x, int y) { if (a[x] == a[y]) { if (b[x] == b[y])return c[x] > c[y]; else return b[x] > b[y]; } else { return a[x] > a[y]; }});
    return ind;
}
template<class T> void sort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; ++i)sort(a[i]); }
template<class T> void rsort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; ++i)rsort(a[i]); }

template<typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) { for(int i=0;i<N;i++)a[i] = v; }
template<typename A, size_t N, size_t O, typename T> void fill(A (&a)[N][O], const T &v) { for(int i=0;i<N;i++)for(int j=0;j<O;j++)a[i][j] = v; }
template<typename A, size_t N, size_t O, size_t P, typename T> void fill(A (&a)[N][O][P], const T &v) { for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)a[i][j][k] = v; }
template<typename A, size_t N, size_t O, size_t P, size_t Q, typename T> void fill(A (&a)[N][O][P][Q], const T &v) { for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)for(int l=0;l<Q;l++)a[i][j][k][l] = v; }
template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, typename T> void fill(A (&a)[N][O][P][Q][R], const T &v) { for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)for(int l=0;l<Q;l++)for(int m=0;m<R;m++)a[i][j][k][l][m] = v; }
template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S, typename T> void fill(A (&a)[N][O][P][Q][R][S], const T &v) { for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)for(int l=0;l<Q;l++)for(int m=0;m<R;m++)for(int n=0;n<S;n++)a[i][j][k][l][m][n] = v; }
template<typename W, typename T> void fill(W &xx, const T vall) { xx = vall; }
template<typename W, typename T> void fill(vector<W> &vecc, const T vall) { for (auto &&vx     : vecc)fill(vx, vall); }
template<typename W, typename T> void fill(vector<W> &xx, const T v, int len) {for(int i=0;i<len;i++) xx[i] = v; }
template<typename W, typename T> void fill(vector<vector<W>> &xx, const T v, int lh, int lw) { for(int i=0;i<lh;i++)for(int j=0;j<lw;j++)xx[i][j] = v; }
template<class T, class U> void fill(vector<T> &a, U val, vi &ind) { fora(v, ind)a[v] = val; }

template<typename A, size_t N> A sum(A (&a)[N]) {
    A res = 0;
    for(int i=0;i<N;i++)res += a[i];
    return res;
}
template<typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {
    A res = 0;
    for(int i=0;i<N;i++)for(int j=0;j<O;j++)res += a[i][j];
    return res;
}
template<typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {
    A res = 0;
    for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)res += a[i][j][k];
    return res;
}
template<typename A, size_t N, size_t O, size_t P, size_t Q> A sum(A (&a)[N][O][P][Q]) {
    A res = 0;
    for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)for(int l=0;l<Q;l++)res += a[i][j][k][l];
    return res;
}
template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A sum(A (&a)[N][O][P][Q][R]) {
    A res = 0;
    for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)for(int l=0;l<Q;l++)for(int m=0;m<R;m++)res += a[i][j][k][l][m];
    return res;
}
template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A sum(A (&a)[N][O][P][Q][R][S]) {
    A res = 0;
    for(int i=0;i<N;i++)for(int j=0;j<O;j++)for(int k=0;k<P;k++)for(int l=0;l<Q;l++)for(int m=0;m<R;m++)for(int n=0;n<S;n++)res += a[i][j][k][l][m][n];
    return res;
}
//@汎用便利関数 入力
int in() {
    int ret;
    cin >> ret;
    return ret;
}
string sin() {
    string ret;
    cin >> ret;
    return ret;
}
template<class T> void in(T &head) { cin >> head; }
template<class T, class... U> void in(T &head, U &... tail) {
    cin >> head;
    in(tail...);
}

#define o_din(o1, o2, o3, o4, o5, o6, name, ...) name
#define din1(a) int a;cin>>a
#define din2(a, b) int a,b;cin>>a>> b
#define din3(a, b, c) int a,b,c;cin>>a>>b>>c
#define din4(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d
#define din5(a, b, c, d, e) int a,b,c,d,e;cin>>a>>b>>c>>d>>e
#define din6(a, b, c, d, e, f) int a,b,c,d,e,f;cin>>a>>b>>c>>d>>e>>f
#define din(...) o_din(__VA_ARGS__,din6,din5,din4,din3,din2 ,din1)(__VA_ARGS__)

#define o_dins(o1, o2, o3, o4, o5, o6, name, ...) name
#define dins1(a) str a;cin>>a
#define dins2(a, b) str a,b;cin>>a>> b
#define dins3(a, b, c) str a,b,c;cin>>a>>b>>c
#define dins4(a, b, c, d) str a,b,c,d;cin>>a>>b>>c>>d
#define dins5(a, b, c, d, e) str a,b,c,d,e;cin>>a>>b>>c>>d>>e
#define dins6(a, b, c, d, e, f) str a,b,c,d,e,f;cin>>a>>b>>c>>d>>e>>f
#define dins(...) o_dins(__VA_ARGS__,dins6,dins5,dins4,dins3,dins2 ,dins1)(__VA_ARGS__)

#define o_dind(o1, o2, o3, o4, name, ...) name
#define din1d(a) din1(a);a--
#define din2d(a, b) din2(a,b);a--,b--
#define din3d(a, b, c) din3(a,b,c);a--,b--,c--
#define din4d(a, b, c, d) din4(a,b,c,d);a--,b--,c--,d--
#define dind(...) o_dind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)

template<class T> void out2(T &&head) { cout << head; }
template<class T, class... U> void out2(T &&head, U &&... tail) {
    cout << head << " ";
    out2(tail...);
}
template<class T, class... U> void out(T &&head, U &&... tail) {
    cout << head << " ";
    out2(tail...);
    cout << "" << endl;
}
template<class T> void out(T &&head) { cout << head << endl; }
void out() { cout << "" << endl; }
#ifdef _DEBUG
template<class T> string out_m(vector<T> &a, int W = inf) {
    stringstream ss;
    if (W == inf)W = min(sz(a), 12ll);
    if (sz(a) == 0)return ss.str();
    for(int i=0;i<W;i++) { ss << a[i] << " "; }
    ss << "" << endl;
    return ss.str();
}
template<class T> string out_m(vector<vector<T> > &a, int H = inf, int W = inf, int key = -1) {
    H = min({H, sz(a), 12ll});
    W = min({W, sz(a[0]), 12ll});
    stringstream ss;
    ss << endl;
    if (key == -1)ss << " *|"; else ss << " " << key << "|";
    for(int w=0;w<W;w++)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    for(int w=0;w<W*4+3;w++)ss << "_";
    ss << "" << endl;
    for(int h=0;h<H;h++) {
        ss << std::right << std::setw(2) << h << "|";
        for(int w=0,lim=min(a[h].size(),12ll);w<W;w++){ if (a[h][w] == linf) ss << "  e" << " "; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    ss << endl;
    return ss.str();
}
/*@formatter:off*/
template<class T> string out_m(vector<vector<vector<T> > > &a, int H = inf, int W = inf, int U = inf) {stringstream ss;    if (H == inf)H = 5;    H = min(H, sz(a));    for(int i=0;i<H;i++) {        ss << endl;        ss << out_m(a[i], W, U, i);    }    ss << endl;    return ss.str();}
string out_m(int a) {stringstream ss;ss << a << endl;return ss.str();}
template<class T> string out_m(T &a) {stringstream ss;ss << a << endl;return ss.str();}
template<class T> void outv(vector<T> &a, int W=inf) {cout << out_m(a,W) << endl;}
template<class T> void outv(vector<vector<T> > &a, int H = linf, int W = linf,int key=-1) {    cout << out_m(a,H,W,key) << endl;}
template<class T> void outv(vector<vector<vector<T> > > &a, int H = linf, int W = linf,int U = linf) {cout << out_m(a,H,W,U)<< endl;}
#else
template<class T> void outv(vector<T> &a, int W = inf) {
    for(int i=0,lim=min(W,(int)a.size());i<lim;i++)  { cout << a[i] << " "; }
    cout << "" << endl;
}
template<class T> void outv(vector<vector<T> > &a, int H = linf, int W = linf, int key = -1) { for(int i=0,lim=min(H,(int)a.size());i<lim;i++){ outv(a[i], W); }}
template<class T> void outv(vector<vector<vector<T> > > &a, int H = linf, int W = linf, int U = linf) { ; }
#endif
template<class T> void outl(vector<T> &a, int n = inf) {  for(int i=0,lim=min(n,(int)a.size());i<lim;i++)cout << a[i] << endl; }
//テーブルをスペースなしで出力
template<class T> void outt(vector<vector<T>>&a){     for(int i=0,lim=(int)a.size();i<lim;i++){        for(int j=0,limj=(int)a[i].size();j<limj;j++){            cout<<a[i][j];        }        cout<<endl;    }}
//int型をbit表記で出力
void outb(int a){cout << bitset<20>(a) << endl;}
template<class T> void na(vector<T> &a, int n) {    a.resize(n);    for(int i=0;i<n;i++)cin >> a[i];}
#define dna(a, n) vector<int> a(n); for(int dnai=0;dnai<n;dnai++) cin >> a[dnai];
#define dnad(a, n) vector<int> a(n); for(int dnai=0;dnai<n;dnai++) cin >> a[dnai],a[dnai]--;
template<class T> void nao(vector<T> &a, int n) {    a.resize(n + 1);    a[0] = 0;     for(int i=0;i<n;i++)cin >> a[i + 1];}
template<class T> void naod(vector<T> &a, int n) {    a.resize(n + 1);    a[0] = 0;    for(int i=0;i<n;i++)cin >> a[i + 1], a[i + 1]--;}
template<class T> void nad(vector<T> &a, int n) {    a.resize(n);    for(int i=0;i<n;i++)cin >> a[i], a[i]--;}
template<class T, class U> void na2(vector<T> &a, vector<U> &b, int n) {    a.resize(n);    b.resize(n);    for(int i=0;i<n;i++)cin >> a[i] >> b[i];}
#define dna2(a, b, n) vi a(n),b(n);for(int dna2i=0;dna2i<n;dna2i++)cin >> a[dna2i] >> b[dna2i];
template<class T, class U> void nao2(vector<T> &a, vector<U> &b, int n) {    a.resize(n + 1);    b.resize(n + 1);    a[0] = b[0] = 0;    for(int i=0;i<n;i++)cin >> a[i + 1] >> b[i + 1];}
#define dna2d(a, b, n) vi a(n),b(n);for(int dna2di=0;dna2di<n;dna2di++){cin >> a[dna2di] >> b[dna2di];a[dna2di]--,b[dna2di]--;}
template<class T, class U> void na2d(vector<T> &a, vector<U> &b, int n) {    a.resize(n);    b.resize(n);    for(int i=0;i<n;i++)cin >> a[i] >> b[i], a[i]--, b[i]--;}
template<class T, class U, class W> void na3(vector<T> &a, vector<U> &b, vector<W> &c, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    for(int i=0;i<n;i++)cin >> a[i] >> b[i] >> c[i];}
#define dna3(a, b, c, n) vi a(n),b(n),c(n);   for(int dna3i=0;dna3i<n;dna3i++)cin >> a[dna3i] >> b[dna3i] >> c[dna3i];
template<class T, class U, class W> void na3d(vector<T> &a, vector<U> &b, vector<W> &c, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    for(int i=0;i<n;i++)cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;}
#define dna3d(a, b, c, n) vi a(n),b(n),c(n);  for(int dna3di=0;dna3di<n;dna3di++){cin >> a[dna3di] >> b[dna3di] >> c[dna3di];a[dna3di]--,b[dna3di]--,c[dna3di]--;}
template<class T, class U, class W, class X> void na4(vector<T> &a, vector<U> &b, vector<W> &c, vector<X> &d, int n) {a.resize(n);    b.resize(n);    c.resize(n);    d.resize(n);    for(int i=0;i<n;i++)cin >> a[i] >> b[i] >> c[i] >> d[i];}
#define dna4(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   for(int dna4i=0;dna4i<n;dna4i++)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i];
#define dna4d(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   for(int dna4i=0;dna4i<n;dna4i++)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i],--a[dna4i] ,-- b[dna4i],-- c[dna4i],--d[dna4i];
#define nt(a, h, w) resize(a,h,w);for(int nthi=0; nthi<H; nthi++)for(int ntwi=0; ntwi<W; ntwi++) cin >> a[nthi][ntwi];
#define ntd(a, h, w) resize(a,h,w);for(int ntdhi=0; ntdhi<H; ntdhi++)for(int ntdwi=0; ntdwi<W; ntdwi++) cin >> a[ntdhi][ntdwi], a[ntdhi][ntdwi]--;
#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');for(int ntphi=1; ntphi<H+1; ntphi++)for(int ntpwi=1; ntpwi<W+1; ntpwi++) cin >> a[ntphi][ntpwi];
//デバッグ
#define sp << " " <<

#define debugName(VariableName) # VariableName

#define deb1(x)  debugName(x)<<" = "<<out_m(x)
#define deb2(x, ...) deb1(x) <<", "<< deb1(__VA_ARGS__)
#define deb3(x, ...) deb1(x) <<", "<< deb2(__VA_ARGS__)
#define deb4(x, ...) deb1(x) <<", "<< deb3(__VA_ARGS__)
#define deb5(x, ...) deb1(x) <<", "<< deb4(__VA_ARGS__)
#define deb6(x, ...) deb1(x) <<", "<< deb5(__VA_ARGS__)
#define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)
#define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)
#define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)
#define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)

#define o_ebug(o1, o2, o3, o4, o5, o6, o7, o8, o9, o10, name, ...) name

#ifdef _DEBUG
#define deb(...)  cerr<< o_ebug(__VA_ARGS__,deb10,deb9,deb8,deb7,deb6,deb5,deb4,deb3,deb2,deb1)(__VA_ARGS__) <<endl
#else
#define deb(...) ;
#endif


#define debugline(x) cerr << x << " " << "(L:" << __LINE__ << ")" << '\n'


//@formatter:off
//よく使うクラス、構造体
struct unionfind {
    vector<int> par;
    vector<int> siz;
    vector<int> es;
    int n, trees;//連結グループの数(親の種類)
    unionfind(int n) : n(n), trees(n) {        par.resize(n);        siz.resize(n);        es.resize(n);        for (int i = 0; i < n; i++) {            par[i] = i;            siz[i] = 1;        }    }
    int root(int x) { if (par[x] == x) { return x; } else { return par[x] = root(par[x]); }}
    bool unite(int x, int y) {
        x = root(x);
        y = root(y);
        es[x]++;
        if (x == y) return false;
        if (siz[x] > siz[y]) swap(x, y);
        trees--;
        par[x] = y;
        siz[y] += siz[x];
        es[y] += es[x];
        return true;
    }
    bool same(int x, int y) { return root(x) == root(y); }
    int size(int x) { return siz[root(x)]; }
    int esize(int x) { return es[root(x)]; }
    vi sizes(){        vi cou(n);        vi ret;        ret.reserve(n);        for(int ntdhi=0; ntdhi<H; ntdhi++)for(int i=0; i<n; i++){            cou[root (i)]++;        }       for(int i=0; i<n; i++){            if(cou[i])ret.push_back(cou[i]);        }        return ret;    }
    //つながりを無向グラフと見なし、xが閉路に含まれるか判定
    bool close(int x) { return esize(x) >= size(x); }
    vec<vi> sets() {        vi ind(n, -1);        int i = 0;        vvi(res, trees);       for(int j=0; j<n; j++) {            int r = root(j);            if (ind[r] == -1)ind[r] = i++;            res[ind[r]].push_back(j);        }        for(int i=0; i<trees; i++) {            int r = root(res[i][0]);            if (res[i][0] == r)continue;            for(int j=1,lim=res[i].size(); j<lim; j++) {                if (res[i][j] == r) {                    swap(res[i][0], res[i][j]);                    break;                }            }        }        return res;    }
};//@formatter:off


using bll =__int128;
using u32 = unsigned;
using u64 = unsigned long long;
using u128 = __uint128_t;

std::ostream &operator<<(std::ostream &dest, __int128_t value) {    std::ostream::sentry s(dest);    if (s) {        __uint128_t tmp = value < 0 ? -value : value;        char buffer[128];        char *d = std::end(buffer);        do {            --d;            *d = "0123456789"[tmp % 10];            tmp /= 10;        } while (tmp != 0);        if (value < 0) {            --d;            *d = '-';        }        int len = std::end(buffer) - d;        if (dest.rdbuf()->sputn(d, len) != len) { dest.setstate(std::ios_base::badbit); }    }    return dest;}
//__int128 toi128(string &s) {    __int128 ret = 0;    for (int i = 0; i < s.length(); ++i)        if ('0' <= s[i] && s[i] <= '9')            ret = 10 * ret + s[i] - '0';    return ret;}


//エラー
void ole() {
#ifdef _DEBUG
    debugline("ole");    exit(0);
#endif
    string a = "a";    for(int i=0; i<30; i++)a += a;    for(int i=0,lim=(1<<17); i<lim; i++)cout << a << endl;    cout << "OLE 出力長制限超過" << endl;    exit(0);}
void re() {    assert(0 == 1);    exit(0);}
void tle() { while (inf)cout << inf << endl; }

//便利関数

//テスト用
char ranc() { return (char) ('a' + rand() % 26); }
int rand(int min, int max) {    assert(min <= max);    if (min >= 0 && max >= 0) { return rand() % (max + 1 - min) + min; } else if (max < 0) { return -rand(-max, -min); } else { if (rand() % 2) { return rand(0, max); } else { return -rand(0, -min); }}}
vi ranv(int n, int min, int max) {    vector<int> v(n);    for(int i=0; i<n; i++)v[i] = rand(min, max);    return v;}
str ransu(int n) {    str s;    for(int i=0; i<n; i++)s += (char) rand('A', 'Z');    return s;}
str ransl(int n) {    str s;   for(int i=0; i<n; i++)s += (char) rand('a', 'z');    return s;}
//単調増加
vi ranvinc(int n, int min, int max) {    vi v(n);    bool bad = 1;    while (bad) {        bad = 0;        v.resize(n);        for(int i=0; i<n; i++) {            if (i && min > max - v[i - 1]) {                bad = 1;                break;            }            if (i)v[i] = v[i - 1] + rand(min, max - v[i - 1]); else v[i] = rand(min, max);        }    }    return v;}
//便利 汎用

void ranvlr(int n, int min, int max, vi &l, vi &r) {    l.resize(n);    r.resize(n);    for(int i=0; i<n; i++) {        l[i] = rand(min, max);        r[i] = l[i] + rand(0, max - l[i]);    }}
vp run_length(vi &a) {    vp ret;    ret.eb(a[0], 1);    for(int i=1,lim=a.size(); i<lim; i++) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}    return ret;}
vector<pair<char, int>> run_length(string &a) {    vector<pair<char, int>> ret;    ret.eb(a[0], 1);    for(int i=1,lim=a.size(); i<lim; i++)  { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}    return ret;}
template<class F> int mgr(int ok, int ng, F f) {    if (ok < ng)        while (ng - ok > 1) {            int mid = (ok + ng) / 2;            if (f(mid))ok = mid; else ng = mid;        }    else        while (ok - ng > 1) {            int mid = (ok + ng) / 2;            if (f(mid))ok = mid; else ng = mid;        }    return ok;}
//strを整数として比較
string smax(str &a, str b) {    if (sz(a) < sz(b)) { return b; }    else if (sz(a) > sz(b)) { return a; }    else if (a < b)return b;    else return a;}
//strを整数として比較
string smin(str &a, str b) {    if (sz(a) > sz(b)) { return b; }    else if (sz(a) < sz(b)) { return a; }    else if (a > b)return b;    else return a;}
template<typename W, typename T> int find(vector<W> &a, const T key) {    for(int i=0,lim=a.size(); i<lim; i++) if (a[i] == key)return i;    return -1;}
template<typename W, typename T> P find(vector<vector<W >> &a, const T key) {    for(int i=0,lim=a.size(); i<lim; i++) for(int j=0,limj=a[0].size(); j<limj; j++)if (a[i][j] == key)return mp(i, j);    return mp(-1, -1);}
template<typename W, typename U> T find(vector<vector<vector<W >>> &a, const U key) {   for(int i=0,lim=a.size(); i<lim; i++)for(int j=0,limj=a[0].size(); j<limj; j++)for(int k=0,limk=a[0][0].size(); k<limk; k++)if (a[i][j][k] == key)return mt(i, j, k);    return mt(-1, -1, -1);}


template<typename W, typename T> int count2(W &a, const T k) { return a == k; }
template<typename W, typename T> int count2(vector<W> &a, const T k) {    int ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
template<typename W, typename T> int count(vector<W> &a, const T k) {    int ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
int count(str &a, str k) {    int ret = 0, len = k.length();    auto pos = a.find(k);    while (pos != string::npos)pos = a.find(k, pos + len), ++ret;    return ret;}
vi count(str &a) {    vi cou(26);    char c = 'a';    if ('A' <= a[0] && a[0] <= 'Z')c = 'A';    for(int i=0,lim=a.size(); i<lim; i++)++cou[a[i] - c];    return cou;}
#define couif count_if
//algorythm


int rev(int a) {    int res = 0;    while (a) {        res *= 10;        res += a % 10;        a /= 10;    }    return res;}
template<class T> void rev(vector<T> &a) { reverse(all(a)); }
template<class U> void rev(vector<vector<U>> &a) {    vector<vector<U> > b(sz(a[0]), vector<U>(sz(a)));    for(int h=0,limh=a.size(); h<limh; h++) for(int w=0,limw=a[0].size(); w<limw; w++)b[w][h] = a[h][w];    a = b;}
void  rev(string &a) { reverse(all(a)); }
constexpr long long p10[] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000ll, 100000000000ll, 1000000000000ll, 10000000000000ll, 100000000000000ll, 1000000000000000ll, 10000000000000000ll, 100000000000000000ll, 1000000000000000000ll};

int get(int a, int keta) { return (a / (int) pow(10, keta)) % 10; }
int keta(int v) { if (v < p10[9]) { if (v < p10[4]) { if (v < p10[2]) { if (v < p10[1]) return 1; else return 2; } else { if (v < p10[3]) return 3; else return 4; }} else { if (v < p10[7]) { if (v < p10[5]) return 5; else if (v < p10[6])return 6; else return 7; } else { if (v < p10[8])return 8; else return 9; }}} else { if (v < p10[13]) { if (v < p10[11]) { if (v < p10[10]) return 10; else return 11; } else { if (v < p10[12]) return 12; else return 13; }} else { if (v < p10[15]) { if (v < p10[14]) return 14; else if (v < p10[15])return 15; else return 16; } else { if (v < p10[17])return 17; else return 18; }}}}
int dsum(int v,int sin=10) {    int ret = 0;    for (; v; v /= sin)ret += v % sin;    return ret;}

struct sint {
    int v;
    sint(long long v) : v(v) {}
    operator long long() { return v; }
    //下からi番目
    int operator[](int i) { return (v / p10[i]) % 10; }
    int back(int i) { return operator[](i); }
    //上からi番目
    int top(int i) {
        int len = keta(v);
        return operator[](len - 1 - i);
    }
    //先頭からi番目にセット
    int settop(int i, int k) {
        int len = keta(v);
        return set(len - 1 - i, k);
    }
    int set(int i, int k) {
        if (i < 0)return settop(abs(i) - 1, k);
        return v += p10[i] * (k - (v / p10[i]) % 10);
    }
    int add(int i, int k = 1) { return v += p10[i] * k; }
    int addtop(int i, int k = 1) { return v += p10[keta(v) - i - 1] * k; }
    int dec(int i, int k = 1) { return v -= p10[i] * k; }
    int dectop(int i, int k = 1) { return v -= p10[keta(v) - i - 1] * k; }
#define op(t, o)template<class T> t operator o(T r){return v o r;}
    op(int, +=);    op(int, -=);    op(int, *=);    op(int, /=);    op(int, %=);    op(int, +);    op(int, -);    op(int, *);    op(int, /);    op(int, %);    op(bool, ==);    op(bool, !=);    op(bool, <);    op(bool, <=);    op(bool, >);    op(bool, >=);
#undef op
    template<class T>  int operator<<=(T r) { return v *= p10[r]; }
    template<class T>  int operator<<(T r) { return v * p10[r]; }
    template<class T>  int operator>>=(T r) { return v /= p10[r]; }
    template<class T>  int operator>>(T r) { return v / p10[r]; }
};
long long mask10(long long v) { return p10[v] - 1; }
//変換系
//[v] := iとなるようなvectorを返す
//存在しない物は-1
template<class T> auto keys(T& a) {    vector<decltype((a.begin())->fi)> res;    for (auto &&k :a)res.push_back(k.fi);    return res;}
template<class T> auto values(T& a) {    vector<decltype((a.begin())->se)> res;    for (auto &&k :a)res.push_back(k.se);    return res;}
template<class T, class U>  bool chma(T &a, const U &b) {    if (a < b) {        a = b;        return true;    }    return false;}
template<class T, class U>  bool chmi(T &a, const U &b) {    if (b < a) {        a = b;        return true;    }    return false;}
template<class T>  T min(T a, signed b) { return a < b ? a : b; }
template<class T>  T max(T a, signed b) { return a < b ? b : a; }
template<class T>  T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }
template<class T>  T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }
template<class T>  T min(vector<T>& a) { return *min_element(all(a)); }
template<class T>  T mini(vector<T>& a) { return min_element(all(a)) - a.begin(); }
template<class T>  T min(vector<T>& a, int n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T min(vector<T>& a, int s, int n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a) { return *max_element(all(a)); }
template<class T,class U>  T max(vector<T>& a,vector<U>& b) { return max(*max_element(all(a)),*max_element(all(b))); }
template<class T>  T maxi(vector<T>& a) { return max_element(all(a)) - a.begin(); }
template<class T>  T max(vector<T>& a, int n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a, int s, int n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<typename A, size_t N>  A max(A (&a)[N]) {    A res = a[0];    for(int i=0; i<N; i++)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O>  A max(A (&a)[N][O]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P>  A max(A (&a)[N][O][P]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q>  A max(A (&a)[N][O][P][Q], const T &v) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>  A max(A (&a)[N][O][P][Q][R]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S>  A max(A (&a)[N][O][P][Q][R][S]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}
template<typename A, size_t N>  A min(A (&a)[N]) {    A res = a[0];    for(int i=0; i<N; i++)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O>  A min(A (&a)[N][O]) {    A res = min(a[0]);    rep(i, N)res = min(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P>  A min(A (&a)[N][O][P]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q>  A min(A (&a)[N][O][P][Q], const T &v) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>  A min(A (&a)[N][O][P][Q][R]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S>  A min(A (&a)[N][O][P][Q][R][S]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}
template<class T> T sum(vector<T> &v, int s, int t) {    T ret = 0;    for(int i=s,lim=min((int)v.size(),t); i<lim; i++)ret += v[i];    return ret;}template<class T> T sum(vector<T> &v, int t=inf) {    return sum(v, 0, t);}template<class T> T sum(vector<vector<T> > &v) {    T ret = 0;    rep(i, sz(v))ret += sum(v[i]);    return ret;}template<class T> T sum(vector<vector<vector<T> > > &v) {    T ret = 0;    rep(i, sz(v))ret += sum(v[i]);    return ret;}template<class T> T sum(vector<vector<vector<vector<T> > > > &v) {    T ret = 0;    rep(i, sz(v))ret += sum(v[i]);    return ret;}template<class T> T sum(vector<vector<vector<vector<vector<T> > > > > &v) {    T ret = 0;    rep(i, sz(v))ret += sum(v[i]);    return ret;}template<class T> auto sum(priority_queue<T, vector<T>, greater<T> > &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}template<class T> auto sum(priority_queue<T> &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}
//template<class T, class U, class... W>  auto sumn(vector<T> &v, U head, W... tail) {    auto ret = sum(v[0], tail...);    rep(i, 1, min(sz(v), head))ret += sum(v[i], tail...);    return ret;}
vi v_i(vi &a) {    int n = max(a) + 1;    vi ret(n, -1);    rep(i, sz(a)) {        ret[a[i]] = i;    }    return ret;}
void clear(PQ &q) { q = PQ(); }
template<class T> void clear(queue<T> &q) { while (q.size())q.pop(); }
template<class T> T *negarr(int size) {    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));    return body + size;}
template<class T> T *negarr2(int h, int w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (int i = 1; i <= 2 * h + 1; ++i) { dummy1[i] = dummy1[i - 1] + 2 * w + 1; }    double **a = dummy1 + h;    return a;}
//imoは0-indexed
//ruiは1-indexed
template<class T> vector<T> imo(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];    return ret;}
//kと同じものの数
template<class T, class U> vi imo(vector<T> &a, U k) {vector<T> ret = a;rep(i, sz(ret))ret[i] = a[i] == k;rep(i, sz(ret) - 1)ret[i + 1] += ret[i];return ret;}
template<class T> vector<T> imox(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}
//漸化的に最小を持つ
template<class T> vector<T> imi(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}
template<class T> struct ruiC {    const vector<T> rui;    ruiC(vector<T> &ru) : rui(ru) {}    T operator()(int l, int r) {        if (l > r) {            cerr<<"ruic ";deb(l, r);assert(0);        }        return rui[r] - rui[l];    }    T operator[](int i) { return rui[i]; }    T back() { return rui.back(); }    int size() { return rui.size(); }};
template<class T> struct rruic {    const T *rrui;    rruic(T *ru) : rrui(ru) {}     T operator()(int l, int r) {        assert(l >= r);        return rrui[r] - rrui[l];    }     T operator[](int i) { return rrui[i]; }};
template<class T>ostream &operator<<(ostream &os, ruiC<T> a) {fora(v,a.rui)os<<v<<" ";return os;}
template<class T> vector<T> ruiv(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i];    return ret;}
template<class T> ruiC<T> ruic(vector<T> &a) {    vector<T> ret = ruiv(a);    return ruiC<T>(ret);}
vector<long long> ruiv(string &a) {    if (sz(a) == 0)return vi(1);    long long dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;    vector<long long> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec;    return ret;}
ruiC<long long> ruic(string &a) {    vector<long long> ret = ruiv(a);    return ruiC<long long>(ret);}
//kと同じものの数
template<class T, class U> vi ruiv(T &a, U k) {    vi ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + (a[i] == k);    return ret;}
template<class T, class U> ruiC<long long> ruic(T &a, U k) {    vi ret = ruiv(a, k);    return ruiC<long long>(ret);}
//xor
template<class T> struct ruixC {    const vector<T> rui;    ruixC(vector<T> &ru) : rui(ru) {}    T operator()(long long l, long long r) {        if (l > r) {            cerr << "ruiXc ";            deb(l, r);            assert(0);        }        return rui[r] ^ rui[l];    }    T operator[](long long i) { return rui[i]; }    T back() { return rui.back(); }    long long size() { return rui.size(); }};
template<class T> vector<T> ruix(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] ^ a[i];    return ret;}
template<class T> ruixC<long long> ruixc(vector<T>  &a) {vi ret = ruix(a);return ruixC<long long>(ret);}
template<class T> vector<T> ruim(vector<T> &a) {    vector<T> res(a.size() + 1, 1);    rep(i, a.size())res[i + 1] = res[i] * a[i];    return res;}
//漸化的に最小を1indexで持つ
template<class T> vector<T> ruimi(vector<T> &a) {    long long n = sz(a);    vector<T> ret(n + 1);    rep(i, 1, n) {        ret[i] = a[i - 1];        chmi(ret[i + 1], ret[i]);    }    return ret;}
//template<class T> T *rrui(vector<T> &a) {
//右から左にかけての半開区間 (-1 n-1]
template<class T> rruic<T> rrui(vector<T> &a) {    long long len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    rer(i, len - 1)res[i - 1] = res[i] + a[i];    return rruic<T>(res);}
//掛け算
template<class T> T *rruim(vector<T> &a) {    long long len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    res[len - 1] = 1;    rer(i, len - 1)res[i - 1] = res[i] * a[i];    return res;}
template<class T, class U> void inc(T &a, U v = 1) { a += v; }
template<class T, class U> void inc(vector<T> &a, U v = 1) { for (auto &u:a)inc(u, v); }
template<class T, class U> void dec(T &a, U v = 1) { a -= v; }
template<class T, class U> void dec(vector<T> &a, U v = 1) { for (auto &u :a)dec(u, v); }
template<class U> void dec(string &a, U v = 1) { for (auto &u :a)dec(u, v); }
template<class T> void dec(vector<T> &a) { for (auto &u :a)dec(u, 1); }
template<class T,class U> void dec(vector<T> &a,vector<U> &b) { for (auto &u :a)dec(u, 1);for (auto &u :b)dec(u, 1); }
template<class T,class U,class W> void dec(vector<T> &a,vector<U> &b,vector<W>&c ) { for (auto &u :a)dec(u, 1);for (auto &u :b)dec(u, 1);for (auto &u :c)dec(u, 1); }
bool ins(long long h, long long w, long long H, long long W) { return h >= 0 && w >= 0 && h < H && w < W; }
bool ins(long long l, long long v, long long r) { return l <= v && v < r; }
template<class T> bool ins(vector<T> &a, long long i, long long j = 0) { return ins(0, i, sz(a)) && ins(0, j, sz(a)); }
long long u(long long a) { return a < 0 ? 0 : a; }
template<class T> vector<T> u(const vector<T> &a) {    vector<T> ret = a;    fora(v, ret)v = u(v);    return ret;}
#define MIN(a) numeric_limits<a>::min()
#define MAX(a) numeric_limits<a>::max()

//添え字を返す
template<class F> long long goldd_l(long long left, long long right, F calc) {    double GRATIO = 1.6180339887498948482045868343656;    long long lm = left + (long long) ((right - left) / (GRATIO + 1.0));    long long rm = lm + (long long) ((right - lm) / (GRATIO + 1.0));    long long fl = calc(lm);    long long fr = calc(rm);    while (right - left > 10) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + (long long) ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + (long long) ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    long long minScore = MAX(long long);    long long resIndex = left;    for (long long i = left; i < right + 1; ++i) {        long long score = calc(i);        if (minScore > score) {            minScore = score;            resIndex = i;        }    }    return resIndex;}
template<class F> long long goldt_l(long long left, long long right, F calc) {        double GRATIO = 1.6180339887498948482045868343656;        long long lm = left + (long long) ((right - left) / (GRATIO + 1.0));        long long rm = lm + (long long) ((right - lm) / (GRATIO + 1.0));        long long fl = calc(lm);        long long fr = calc(rm);        while (right - left > 10) {            if (fl > fr) {                right = rm;                rm = lm;                fr = fl;                lm = left + (long long) ((right - left) / (GRATIO + 1.0));                fl = calc(lm);            } else {                left = lm;                lm = rm;                fl = fr;                rm = lm + (long long) ((right - lm) / (GRATIO + 1.0));                fr = calc(rm);            }        }    if (left > right) {        long long l = left;        left = right;        right = l;    }    long long maxScore = MIN(long long);    long long resIndex = left;    for (long long i = left; i < right + 1; ++i) {        long long score = calc(i);        if (maxScore < score) {            maxScore = score;            resIndex = i;        }    }    return resIndex;}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template<class F> dou goldd_d(dou left, dou right, F calc, long long loop = 200) {    dou GRATIO = 1.6180339887498948482045868343656;    dou lm = left + ((right - left) / (GRATIO + 1.0));    dou rm = lm + ((right - lm) / (GRATIO + 1.0));    dou fl = calc(lm);    dou fr = calc(rm);    /*200にすればおそらく大丈夫*/    /*余裕なら300に*/    long long k = 141;    loop++;    while (--loop) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    return left;}
template<class F> dou goldt_d(dou left, dou right, F calc, long long loop = 200) {    double GRATIO = 1.6180339887498948482045868343656;    dou lm = left + ((right - left) / (GRATIO + 1.0));    dou rm = lm + ((right - lm) / (GRATIO + 1.0));    dou fl = calc(lm);    dou fr = calc(rm);    loop++;    while (--loop) {        if (fl > fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    return left;}
//l ~ rを複数の区間に分割し、極致を与えるiを返す time-20 msまで探索
template<class F> long long goldd_ls(long long l, long long r, F calc, long long time = 2000) {    auto lim = milliseconds(time - 20);    long long mini = 0, minv = MAX(long long);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        long long haba = (r - l + k) / k;/*((r-l+1) + k-1) /k*/        long long nl = l;        long long nr = l + haba;        rep(i, k) {            long long ni = goldd_l(nl, nr, calc);            if (chmi(minv, calc(ni))) mini = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return mini;}
template<class F> long long goldt_ls(long long l, long long r, F calc, long long time = 2000) {    auto lim = milliseconds(time - 20);    long long maxi = 0, maxv = MIN(long long);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        long long haba = (r - l + k) / k;/*((r-l+1) + k-1) /k*/        long long nl = l;        long long nr = l + haba;        rep(i, k) {            long long ni = goldt_l(nl, nr, calc);            if (chma(maxv, calc(ni))) maxi = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return maxi;}
template<class F> dou goldd_d_s(dou l, dou r, F calc, long long time = 2000) {    /*20ms余裕を持つ*/    auto lim = milliseconds(time - 20);    dou mini = 0, minv = MAX(dou);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        dou haba = (r - l) / k;        dou nl = l;        dou nr = l + haba;        rep(i, k) {            dou ni = goldd_d(nl, nr, calc);            if (chmi(minv, calc(ni))) mini = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return mini;}
template<class F> dou goldt_d_s(dou l, dou r, F calc, long long time = 2000) {    /*20ms余裕を残している*/    auto lim = milliseconds(time - 20);    dou maxi = 0, maxv = MIN(dou);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        dou haba = (r - l) / k;        dou nl = l;        dou nr = l + haba;        rep(i, k) {            dou ni = goldt_d(nl, nr, calc);            if (chma(maxv, calc(ni))) maxi = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return maxi;}
template<class T> T min(vector<vector<T >> &a) {    T res = MAX(T);    rep(i, a.size())chmi(res, *min_element(all(a[i])));    return res;}
template<class T> T max(vector<vector<T >> &a) {    T res = MIN(T);    rep(i, a.size())chma(res, *max_element(all(a[i])));    return res;}
constexpr bool bget(long long m, long long keta) {
    #ifdef _DEBUG
        assert(keta<=62);//オーバーフロー 1^62までしか扱えない
    #endif
    return (m >> keta) & 1; }
long long bget(long long m, long long keta, long long sinsuu) {    m /= (long long) pow(sinsuu, keta);    return m % sinsuu;}
constexpr long long bit(long long n) {
    #ifdef _DEBUG
        assert(n<=62);//オーバーフロー 1^62までしか扱えない
    #endif
    return (1LL << (n)); }
long long bit(long long n, long long sinsuu) { return (long long) pow(sinsuu, n); }
long long mask(long long n) { return (1ll << n) - 1; }
//aをbitに置きなおす
long long bit(vi&a){    int m=0;    for(auto&& v:a) m|=bit(v);    return m;}
#define bcou __builtin_popcountll
//最下位ビット
long long lbit(long long n) { return n & -n; }
long long lbiti(long long n) { return log2(n & -n); }
//最上位ビット
long long hbit(long long n) {    n |= (n >> 1);    n |= (n >> 2);    n |= (n >> 4);    n |= (n >> 8);    n |= (n >> 16);    n |= (n >> 32);    return n - (n >> 1);}
long long hbiti(long long n){return log2(hbit(n));}
long long hbitk(long long n) {    long long k = 0;    rer(i, 5) {        long long a = k + (1ll << i);        long long b = 1ll << a;        if (b <= n)k += 1ll << i;    }    return k;}
//初期化は0を渡す
long long nextComb(long long &mask, long long n, long long r) {    if (!mask)return mask = (1LL << r) - 1;    long long x = mask & -mask; /*最下位の1*/    long long y = mask + x; /*連続した下の1を繰り上がらせる*/    long long res = ((mask & ~y) / x >> 1) | y;    if (bget(res, n))return mask = 0; else return mask = res;}
//n桁以下でビットがr個立っているもののvectorを返す
vi bitCombList(long long n, long long r) {    vi res;    long long m = 0;    while (nextComb(m, n, r)) { res.push_back(m); }    return res;}
//masの立ってるindexを見る
#define forbit(i, mas) for (int forbitj = lbit(mas), forbitm = mas, i = log2(forbitj); forbitm; forbitm = forbitj ? forbitm ^ forbitj : 0, forbitj = lbit(forbitm), i = log2(forbitj))

char itoal(long long i) { return 'a' + i; }
char itoaL(long long i) { return 'A' + i; }
long long altoi(char c) {    if ('A' <= c && c <= 'Z')return c - 'A';    return c - 'a';}
long long ctoi(char c) { return c - '0'; }
char itoc(long long i) { return i + '0'; }
long long vtoi(vi &v) {    long long res = 0;    if (sz(v) > 18) {        debugline("vtoi");        deb(sz(v));        ole();    }    rep(i, sz(v)) {        res *= 10;        res += v[i];    }    return res;}
vi itov(long long i) {    vi res;    while (i) {        res.push_back(i % 10);        i /= 10;    }    rev(res);    return res;}
vi stov(string &a) {    long long n = sz(a);    vi ret(n);    rep(i, n) { ret[i] = a[i] - '0'; }    return ret;}
//基準を満たさないものは0になる
vi stov(string &a, char one) {    long long n = sz(a);    vi ret(n);    rep(i, n)ret[i] = a[i] == one;    return ret;}
vector<vector<long long>> ctoi(vector<vector<char>> s, char c) {    long long n = sz(s), m = sz(s[0]);    vector<vector<long long>> res(n, vector<long long>(m));    rep(i, n)rep(j, m)res[i][j] = s[i][j] == c;    return res;}
#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );
//[i] := vを返す
//aは0~n-1で置き換えられる
vi compress(vi &a) {    vi b;    long long len = a.size();    for (long long i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (long long i = 0; i < len; ++i) { a[i] = lower_bound(all(b), a[i]) - b.begin(); }    long long blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, umap<long long, long long> &map) {    vi b;    long long len = a.size();    for (long long i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (long long i = 0; i < len; ++i) {        long long v = a[i];        a[i] = lower_bound(all(b), a[i]) - b.begin();        map[v] = a[i];    }    long long blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, vi &r) {    vi b;    long long len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    sort(b);    unique(b);    for (long long i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (ll i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    long long blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, vi &r, vi &s) {    vi b;    ll len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    fora(v, s)b.push_back(v);    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (ll i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    for (ll i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vector<vi> &a) {    vi b;    fora(vv, a)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vv, a)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    long long blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vector<vector<vi >> &a) {    vi b;    fora(vvv, a)fora(vv, vvv)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    long long blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
void compress(long long a[], long long len) {    vi b;    for (long long i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (long long i = 0; i < len; ++i) { a[i] = lower_bound(all(b), a[i]) - b.begin(); }}
//要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(all(a),v))
#define lowerIndex(a, v) (lower_bound(all(a),v)-a.begin())
#define lowerBound(a, v) (*lower_bound(all(a),v))
#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())
#define upperBound(a, v) (*upper_bound(all(a),v))
#define rlowerIndex(a, v) (upper_bound(all(a),v)-a.begin()-1)
#define rlowerBound(a, v) *(--(upper_bound(all(a),v)))
#define rupperIndex(a, v) (lower_bound(all(a),v)-a.begin()-1)
#define rupperBound(a, v) *(--(lower_bound(all(a),v)))
#define next2(a) next(next(a))
#define prev2(a) prev(prev(a))

//狭義の単調増加列 長さを返す
template<class T> int lis(vector<T> &a) {    int n = sz(a);    vi tail(n + 1, MAX(T));    rep(i, n) {        int id = lowerIndex(tail, a[i]);/**/        tail[id] = a[i];    }    return lowerIndex(tail, MAX(T));}
template<class T> int lis_eq(vector<T> &a) {    int n = sz(a);    vi tail(n + 1, MAX(T));    rep(i, n) {        int id = upperIndex(tail, a[i]);/**/        tail[id] = a[i];    }    return lowerIndex(tail, MAX(T));}

//iteratorを返す
//valueが1以上の物を返す 0は見つけ次第削除
//vを減らす場合 (*it).se--でいい
template<class T, class U, class V> auto lower_map(map<T, U> &m, V k) {    auto ret = m.lower_bound(k);    while (ret != m.end() && (*ret).second == 0) {        ret = m.erase(ret);    }    return ret;}
template<class T, class U, class V> auto upper_map(map<T, U> &m, V k) {    auto ret = m.upper_bound(k);    while (ret != m.end() && (*ret).second == 0) {        ret = m.erase(ret);    }    return ret;}
//存在しなければエラー
template<class T, class U, class V> auto rlower_map(map<T, U> &m, V k) {    auto ret = upper_map(m, k);    assert(ret != m.begin());    ret--;    while (1) {        if ((*ret).second != 0)break;        assert(ret != m.begin());        auto next = ret;        --next;        m.erase(ret);        ret = next;    }    return ret;}
template<class T, class U, class V> auto rupper_map(map<T, U> &m, V k) {    auto ret = lower_map(m, k);    assert(ret != m.begin());    ret--;    while (1) {        if ((*ret).second != 0)break;        assert(ret != m.begin());        auto next = ret;        --next;        m.erase(ret);        ret = next;    }    return ret;}

template<class T> void fin(T s) { cout << s << endl, exit(0); }

//便利 数学 math
long long mod(long long a, long long m) { return (a % m + m) % m; }
long long pow(long long a) { return a * a; };
long long fact(long long v) { return v <= 1 ? 1 : v * fact(v - 1); }
dou factd(int v){static vd fact(2,1);    if(sz(fact)<=v){        rep(i,sz(fact),v+1){            fact.push_back(fact.back()*i);        }    }    return fact[v];}

long long comi(long long n, long long r) {    assert(n < 100);    static vvi(pas, 100, 100);    if (pas[0][0])return pas[n][r];    pas[0][0] = 1;    rep(i, 1, 100) {        pas[i][0] = 1;        rep(j, 1, i + 1)pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];    }    return pas[n][r];}
double comd2(long long n, long long r) {    static vvd(comb, 2020, 2020);    if (comb[0][0] == 0) {        comb[0][0] = 1;        rep(i, 2000) {            comb[i + 1][0] = 1;            rep(j, 1, i + 2) { comb[i + 1][j] = comb[i][j] + comb[i][j - 1]; }        }    }    return comb[n][r];}
double comd(int n, int r) {    if (r < 0 || r > n) return 0;    if (n < 2020)return comd2(n, r);    static vd fact(2, 1);    if (sz(fact) <= n) { rep(i, sz(fact), n + 1) { fact.push_back(fact.back() * i); }}    return fact[n] / fact[n - r] / fact[r];}

long long gcd(long long a, long long b) {while (b) a %= b, swap(a, b);return abs(a);}
long long gcd(vi b) {long long res = b[0];rep(i, 1, sz(b))res = gcd(b[i], res);return res;}
long long lcm(long long a, long long b) { return a / gcd(a, b) * b; }
long long lcm(vi a) {long long res = a[0];rep(i, 1, sz(a))res = lcm(a[i], res);return res;}
long long ceil(long long a, long long b) {if (b == 0) {debugline("ceil");deb(a, b);ole();return -1;} else if (a < 0) { return 0; } else { return (a + b - 1) / b; }}
long long lower_remi__bx_a(long long kei, long long rem, long long x) {if (rem >= x) return 0;return (x - rem + kei - 1) / kei;}
long long lower_remv__bx_a(long long kei, long long rem, long long x) {if (rem >= x) return rem;return (x - rem + kei - 1) / kei * kei + rem;}
long long upper_remi__bx_a(long long kei, long long rem, long long x) {if (rem > x) return 0;return (x - rem + kei) / kei;}
long long upper_remv__bx_a(long long kei, long long rem, long long x) {if (rem > x) return rem;return (x - rem + kei) / kei * kei + rem;}
//v * v >= aとなる最小のvを返す
long long sqrt(long long a) {if (a < 0) {debugline("sqrt");deb(a);ole();}long long res = (long long) std::sqrt(a);while (res * res < a)++res;return res;}
double log(double e, double x) { return log(x) / log(e); }
long long sig(long long t) { return ((1 + t) * t) >> 1; }
long long sig(long long s, long long t) { return ((s + t) * (t - s + 1)) >> 1; }


//機能拡張
template<class T, class U> void operator+=(pair<T,U> &a, pair<T,U> & b) {a.fi+=b.fi;a.se+=b.se;}
template<class T, class U> pair<T,U> operator+(pair<T,U> &a, pair<T,U> & b) {return pair<T,U>(a.fi+b.fi,a.se+b.se);}

template<typename CharT, typename Traits, typename Alloc>basic_string<CharT, Traits, Alloc>operator+(const basic_string<CharT, Traits, Alloc> &lhs, const int rv) {    basic_string<CharT, Traits, Alloc> str(lhs);    str.append(to_string(rv));    return str;}template<typename CharT, typename Traits, typename Alloc>void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const int rv) {    lhs += to_string(rv);}template<typename CharT, typename Traits, typename Alloc>basic_string<CharT, Traits, Alloc>operator+(const basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {    basic_string<CharT, Traits, Alloc> str(lhs);    str.append(to_string(rv));    return str;}template<typename CharT, typename Traits, typename Alloc>void operator+=(basic_string<CharT, Traits, Alloc> &lhs, const signed rv) {    lhs += to_string(rv);}
template<typename CharT, typename Traits, typename Alloc> void operator*=(basic_string<CharT, Traits, Alloc> &s, int num) {auto bek = s;s = "";for (; num; num >>= 1) {if (num & 1) {s += bek;}bek += bek;}}
template<class T, class U> void operator+=(queue<T> &a, U v) { a.push(v); }template<class T, class U> void operator+=(deque<T> &a, U v) { a.push_back(v); }template<class T> priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, vector <T> &v) {    fora(d, v)a.push(d);    return a;}template<class T, class U> priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, U v) {    a.push(v);    return a;}template<class T, class U> priority_queue<T> &operator+=(priority_queue<T> &a, U v) {    a.push(v);    return a;}template<class T> set<T> &operator+=(set<T> &a, vector <T> v) {    fora(d, v)a.insert(d);    return a;}template<class T, class U> auto operator+=(set<T> &a, U v) { return a.insert(v); }template<class T, class U> auto operator-=(set<T> &a, U v) { return a.erase(v); }template<class T, class U> auto operator+=(mset<T> &a, U v) { return a.insert(v); }template<class T, class U> set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {    a.insert(v);    return a;}template<class T, class U> vector<T> &operator+=(vector<T> &a, U v) {    a.push_back(v);    return a;}template<class T, class U> vector<T> operator+(const vector <T> &a, U v) {    vector<T> ret = a;    ret += v;    return ret;}template<class T, class U> vector<T> operator+(U v, const vector <T> &a) {    vector<T> ret = a;    ret.insert(ret.begin(), v);    return ret;}template<class T> vector<T> operator+(vector<T> a, vector <T> b) {    vector<T> ret;    ret = a;    fora(v, b)ret += v;    return ret;}template<class T> vector<T> &operator+=(vector<T> &a, vector <T> &b) {    rep(i, sz(b)) {/*こうしないとa+=aで両辺が増え続けてバグる*/        a.push_back(b[i]); }    return a;}
template<class T> vector<T> &operator-=(vector<T> &a, vector <T> &b) {    if (sz(a) != sz(b)) {        debugline("vector<T> operator-=");        deb(a);        deb(b);        exit(0);    }    rep(i, sz(a))a[i] -= b[i];    return a;}
template<class T> vector<T> operator-(vector<T> &a, vector<T> &b) {    if (sz(a) != sz(b)) {        debugline("vector<T> operator-");        deb(a);        deb(b);        ole();    }    vector<T> res(sz(a));    rep(i, sz(a))res[i] = a[i] - b[i];    return res;}
template<class T, class U> void operator*=(vector<T> &a, U b) {    vector<T> ta = a;    rep(b-1){        a+=ta;    }}
template<typename T> void erase(vector<T> &v, unsigned long long i) { v.erase(v.begin() + i); }
template<typename T> void erase(vector<T> &v, unsigned long long s, unsigned long long e) { v.erase(v.begin() + s, v.begin() + e); }
template<class T, class U> void erase(map<T, U> &m, long long okl, long long ngr) { m.erase(m.lower_bound(okl), m.lower_bound(ngr)); }
template<class T> void erase(set<T> &m, long long okl, long long ngr) { m.erase(m.lower_bound(okl), m.lower_bound(ngr)); }
template<typename T> void erasen(vector<T> &v, unsigned long long s, unsigned long long n) { v.erase(v.begin() + s, v.begin() + s + n); }
template<typename T, typename U> void insert(vector<T> &v, unsigned long long i, U t) { v.insert(v.begin() + i, t); }
template<typename T, typename U> void push_front(vector<T> &v, U t) { v.insert(v.begin(), t); }
template<typename T, typename U> void insert(vector<T> &v, unsigned long long i, vector<T> list) { for (auto &&va:list)v.insert(v.begin() + i++, va); }
template<typename T> void insert(set<T> &v, vector<T> list) { for (auto &&va :list)v.insert(va); }
vector<string> split(const string a, const char deli) {    string b = a + deli;    long long l = 0, r = 0, n = b.size();    vector<string> res;    rep(i, n) {        if (b[i] == deli) {            r = i;            if (l < r)res.push_back(b.substr(l, r - l));            l = i + 1;        }    }    return res;}
vector<string> split(const string a, const string deli) {    vector<string> res;    long long kn = sz(deli);    std::string::size_type Pos(a.find(deli));    long long l = 0;    while (Pos != std::string::npos) {        if (Pos - l)res.push_back(a.substr(l, Pos - l));        l = Pos + kn;        Pos = a.find(deli, Pos + kn);    }    if (sz(a) - l)res.push_back(a.substr(l, sz(a) - l));    return res;}
void yn(bool a) { if (a)cout << "yes" << endl; else cout << "no" << endl; }
void Yn(bool a) { if (a)cout << "Yes" << endl; else cout << "No" << endl; }
void YN(bool a) { if (a)cout << "YES" << endl; else cout << "NO" << endl; }
void fyn(bool a) {    if (a)cout << "yes" << endl; else cout << "no" << endl;    exit(0);}
void fYn(bool a) {    if (a)cout << "Yes" << endl; else cout << "No" << endl;    exit(0);}
void fYN(bool a) {    if (a)cout << "YES" << endl; else cout << "NO" << endl;    exit(0);}
void Possible(bool a) {    if (a)cout << "Possible" << endl; else cout << "Impossible" << endl;    exit(0);}
void POSSIBLE(bool a) {    if (a)cout << "POSSIBLE" << endl; else cout << "IMPOSSIBLE" << endl;    exit(0);}
template<typename T> class fixed_point        : T {public:    explicit constexpr fixed_point(T &&t) noexcept: T(std::forward<T>(t)) {}    template<typename... Args> constexpr decltype(auto) operator()(Args &&... args) const { return T::operator()(*this, std::forward<Args>(args)...); }};template<typename T> static inline constexpr decltype(auto) fix(T &&t) noexcept { return fixed_point<T>{std::forward<T>(t)}; }


//@起動時
struct initon {
    initon() {
        cin.tie(0);
        ios::sync_with_stdio(false);
        cout.setf(ios::fixed);
        cout.precision(16);
        srand((unsigned) clock() + (unsigned) time(NULL));
    };
} initonv;//@formatter:on

//gra mll pr
//上下左右
const string udlr = "udlr";
string UDLR = "UDLR";//x4と連動 UDLR.find('U') := x4[0]
//右、上が正
constexpr int y4[] = {1, -1, 0, 0};
constexpr int x4[] = {0, 0, -1, 1};
constexpr int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};
constexpr int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};
//ll k, K, d, x, y, z, q, Q,cou;vvi (s, 0, 0);

//ll n, m, k, H, W;
//vi a, b;
//vvc (ba, 0, 0);

int N, H, W;
vi A, B, C;


void solve() {
    int cou=0;N=11;
    rep(i, N){
        cou+=i;
    }
    cout << cou << endl;
}

auto my(int n, vi &a) {
    return 0;
}

auto sister(int n, vi &a) {
    ll ret = 0;
    return ret;
}

signed main() {
    solve();

#define arg n,a
#ifdef _DEBUG
    bool bad = 0;
    for (int i = 0, ok = 1; i < k5 && ok; ++i) {
        int n = rand(1, 8);
        vi a = ranv(n, 1, 10);
        auto myres = my(arg);
        auto res = sister(arg);
        ok = myres == res;
        if (!ok) {
            out(arg);
            cerr << "AC : " << res << endl;
            cerr << "MY  : " << myres << endl;
            bad = 1;


            break;
        }
    }
    if (!bad) {
//        cout << "完璧 : solveを書き直そう" << endl;
//        cout << "     : そして、solve()を呼び出すのだ" << endl;
//        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;
    }
#endif
    return 0;
};


Compiler output:
a.cc: In member function 'vi unionfind::sizes()':
a.cc:981:103: error: 'H' was not declared in this scope
  981 |     vi sizes(){        vi cou(n);        vi ret;        ret.reserve(n);        for(int ntdhi=0; ntdhi<H; ntdhi++)for(int i=0; i<n; i++){            cou[root (i)]++;        }       for(int i=0; i<n; i++){            if(cou[i])ret.push_back(cou[i]);        }        return ret;    }
      |                                                                                                       ^