//#pragma GCC optimize ("-O3")
#include <bits/stdc++.h>
using namespace std;
//衝突対策
#define ws wszzzz
#define int long long

//@formatter:off
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;    }    int operator[](int i) {        assert(i < 4);        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;    }};
typedef T2<int, int, int> T;
typedef F2<int, int, int, int> F;
T mt(int a, int b, int c) {return T(a, b, c);}

//@マクロ省略系 型,構造
#define ll long long
#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<ll, ll>;
using pd =pair<dou, dou>;
#define fi first
#define se second
#define vec vector
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==
#define elf else if

#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))

//マクロ省略系 コンテナ

using vi = vector<int>;
#define _overloadvvi(_1, _2, _3, _4, name, ...) name
#define vvi0() vec<vi>
#define vvi1(a) vec<vi> a
#define vvi2(a, b) vec<vi> a(b)
#define vvi3(a, b, c) vec<vi> a(b,vi(c))
#define vvi4(a, b, c, d) vec<vi> a(b,vi(c,d))
#define vvi(...) _overloadvvi(__VA_ARGS__,vvi4,vvi3,vvi2 ,vvi1,vvi0)(__VA_ARGS__)

using vl = vector<ll>;
#define _overloadvvl(_1, _2, _3, _4, name, ...) name
#define vvl1(a) vec<vl> a
#define vvl2(a, b) vec<vl> a(b)
#define vvl3(a, b, c) vec<vl> a(b,vl(c))
#define vvl4(a, b, c, d) vec<vl> a(b,vl(c,d))
#define vvl(...) _overloadvvl(__VA_ARGS__,vvl4,vvl3,vvl2 ,vvl1)(__VA_ARGS__)

using vb = vector<bool>;
#define _overloadvvb(_1, _2, _3, _4, name, ...) name
#define vvb1(a) vec<vb> a
#define vvb2(a, b) vec<vb> a(b)
#define vvb3(a, b, c) vec<vb> a(b,vb(c))
#define vvb4(a, b, c, d) vec<vb> a(b,vb(c,d))
#define vvb(...) _overloadvvb(__VA_ARGS__,vvb4,vvb3,vvb2 ,vvb1)(__VA_ARGS__)

using vs = vector<string>;
#define _overloadvvs(_1, _2, _3, _4, name, ...) name
#define vvs1(a) vec<vs> a
#define vvs2(a, b) vec<vs> a(b)
#define vvs3(a, b, c) vec<vs> a(b,vs(c))
#define vvs4(a, b, c, d) vec<vs> a(b,vs(c,d))
#define vvs(...) _overloadvvs(__VA_ARGS__,vvs4,vvs3,vvs2 ,vvs1)(__VA_ARGS__)

using vd = vector<double>;
#define _overloadvvd(_1, _2, _3, _4, name, ...) name
#define vvd1(a) vec<vd> a
#define vvd2(a, b) vec<vd> a(b)
#define vvd3(a, b, c) vec<vd> a(b,vd(c))
#define vvd4(a, b, c, d) vec<vd> a(b,vd(c,d))
#define vvd(...) _overloadvvd(__VA_ARGS__,vvd4,vvd3,vvd2 ,vvd1)(__VA_ARGS__)

using vc=vector<char>;
#define _overloadvvc(_1, _2, _3, _4, name, ...) name
#define vvc1(a) vec<vc> a
#define vvc2(a, b) vec<vc> a(b)
#define vvc3(a, b, c) vec<vc> a(b,vc(c))
#define vvc4(a, b, c, d) vec<vc> a(b,vc(c,d))
#define vvc(...) _overloadvvc(__VA_ARGS__,vvc4,vvc3,vvc2 ,vvc1)(__VA_ARGS__)

using vp = vector<P>;
#define _overloadvvp(_1, _2, _3, _4, name, ...) name
#define vvp1(a) vec<vp> a
#define vvp2(a, b) vec<vp> a(b)
#define vvp3(a, b, c) vec<vp> a(b,vp(c))
#define vvp4(a, b, c, d) vec<vp> a(b,vp(c,d))
#define vvp(...) _overloadvvp(__VA_ARGS__,vvp4,vvp3,vvp2 ,vvp1)(__VA_ARGS__)

using vt = vector<T>;
#define _overloadvvt(_1, _2, _3, _4, name, ...) name
#define vvt1(a) vec<vt> a
#define vvt2(a, b) vec<vt> a(b)
#define vvt3(a, b, c) vec<vt> a(b,vt(c))
#define vvt4(a, b, c, d) vec<vt> a(b,vt(c,d))
#define vvt(...) _overloadvvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1)(__VA_ARGS__)

#define v3i(a, b, c, d) vector<vector<vi>> a(b, vector<vi>(c, vi(d)))
#define v3d(a, b, c, d) vector<vector<vd>> a(b, vector<vd>(c, vd(d)))
#define v3m(a, b, c, d) vector<vector<vm>> a(b, vector<vm>(c, vm(d)))

#define _vvi vector<vi>
#define _vvl vector<vl>
#define _vvb vector<vb>
#define _vvs vector<vs>
#define _vvd vector<vd>
#define _vvc vector<vc>
#define _vvp vector<vp>

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<int>(__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<ll, vector<ll>, greater<ll> >
#define tos to_string
using mapi = map<int, int>;
using mapp = map<P, int>;
using mapd = map<dou, int>;
using mapc = map<char, int>;
using maps = map<str, int>;
using seti = set<int>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<int>;
#define bset bitset
#define uset unordered_set
#define mset multiset
#define umap unordered_map
#define umapi unordered_map<int,int>
#define umapp unordered_map<P,int>
#define mmap multimap

//マクロ 繰り返し
#define _overloadrep(_1, _2, _3, _4, name, ...) name
# define rep1(n) for(int rep1i = 0,_lim=n; rep1i < _lim ; rep1i++)
# define _rep(i, n) for(int i = 0,_lim=n; i < _lim ; i++)
#define repi(i, m, n) for(int i = m,_lim=n; i < _lim ; i++)
#define repadd(i, m, n, ad) for(int i = m,_lim=n; i < _lim ; i+= ad)
#define rep(...) _overloadrep(__VA_ARGS__,repadd,repi,_rep,rep1)(__VA_ARGS__)
#define _rer(i, n) for(int i = n; i >= 0 ; i--)
#define reri(i, m, n) for(int i = m,_lim=n; i >= _lim ; i--)
#define rerdec(i, m, n, dec) for(int i = m,_lim=n; i >= _lim ; i-=dec)
#define rer(...) _overloadrep(__VA_ARGS__,rerdec,reri,_rer,)(__VA_ARGS__)

#define reps2(i, j, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)
#define reps3(i, j, k, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)for(int k = 0; k < _lim ; k++)
#define reps4(i, j, k, l, n) for(int i = 0,_lim=n; i < _lim ; i++)for(int j = 0; j < _lim ; j++)for(int k = 0; k < _lim ; k++)for(int l = 0; l < _lim ; l++)

#define _overloadreps(_1, _2, _3, _4, _5, name, ...) name
#define reps(...) _overloadreps(__VA_ARGS__,reps4,reps3,reps2,_rep,)(__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)
#define forg(gi, ve) for (int gi = 0,_lim = ve.size(), f, t, c; gi < _lim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); gi++)
#define fort(gi, ve) for (int gi = 0, f, t, c; gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); gi++)if(t!=p)

#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; it++)

//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 ll linf = (ll) 1e18 + 100;
const char infc = '{';
const string infs = "{";
const double eps = 1e-9;
const double PI = 3.1415926535897932384626433832795029L;
ll ma = numeric_limits<ll>::min();
ll mi = numeric_limits<ll>::max();
const int y4[] = {-1, 1, 0, 0};
const int x4[] = {0, 0, -1, 1};
const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};
const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};

//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 push_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<ll>()(((ll) x.first << 32) + x.second);}};template<> class hash<std::pair<ll, ll>> {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<ll, ll> &x) const {return hash<ll>()(((ll) 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 << endl;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> ostream &operator<<(ostream &os, vector <vector<T>> &vec) {    for (int i = 0; i < vec.size(); i++) {        for (int j = 0; j < vec[0].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<typename V, typename H> void resize(vector<V> &vec, const H head) { vec.resize(head); }template<typename V, typename H, typename ... T> void resize(vector<V> &vec, const H &head, const T ... tail) {vec.resize(head);for (auto &v: vec)resize(v, tail...);}
template<typename T, typename _Pr> bool all_of(const vector<T> &vec, _Pr pred) { return std::all_of(vec.begin(), vec.end(), pred); }
template<typename T, typename _Pr> bool any_of(const vector<T> &vec, _Pr pred) { return std::any_of(vec.begin(), vec.end(), pred); }
template<typename T, typename _Pr> bool none_of(const vector<T> &vec, _Pr pred) { return std::none_of(vec.begin(), vec.end(), pred); }
template<typename T, typename _Pr> const typename vector<T>::const_iterator find_if(const vector<T> &vec, _Pr pred) { return std::find_if(vec.begin(), vec.end(), pred); }
template<class T> bool contains(const std::string &s, const T &v) { return s.find(v) != std::string::npos; }
template<typename T> bool contains(const vector<T> &vec, const T &val) { return std::find(vec.begin(), vec.end(), val) != vec.end(); }
template<typename T, typename _Pr> bool contains_if(const vector<T> &vec, _Pr pred) { return std::find_if(vec.begin(), vec.end(), pred) != vec.end(); }

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, str key, str v) {if (v == "")a.erase(remove(all(a), key[0]), a.end());else replace(all(a), key[0], v[0]);}
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) { rep(i, sz(a)) a[i] = a[i] == k; }
template<class T, class U> void replace(vector<vector<T >> &a, U k) { rep(i, sz(a))rep(j, sz(a[0])) a[i][j] = a[i][j] == k; }
template<class T> void replace(T &a) { replace(a, '#'); }

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;    rep(i, sz(a))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));    rep(i, sz(p))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;    rep(i, sz(a))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;    rep(i, sz(a))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)); }
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);    rep(i, sz(a)) 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);    rep(i, sz(a)) 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);    rep(i, sz(a))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);    rep(i, sz(a))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);    rep(i, sz(a)) 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);    rep(i, sz(a)) 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);    rep(i, sz(a)) 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);    rep(i, sz(a)) a[i] = d[i].f, b[i] = d[i].s, c[i] = d[i].t;}
//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);    rep(i, sz(a))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) { rep(i, N)a[i] = v; }template<typename A, size_t N, size_t O, typename T> void fill(A (&a)[N][O], const T &v) { rep(i, N)rep(j, O)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) { rep(i, N)rep(j, O)rep(k, P)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) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)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) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)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) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)a[i][j][k][l][m][n] = v; }
template<typename V, typename T>void fill(V &xx, const T vall) {    xx = vall;}template<typename V, typename T>void fill(vector<V> &vecc, const T vall) {    for (auto &&vx     : vecc)fill(vx, vall);}
template<typename A, size_t N> A sum(A (&a)[N]) {    A res = 0;    rep(i, N)res += a[i];    return res;}template<typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {    A res = 0;    rep(i, N)rep(j, O)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;    rep(i, N)rep(j, O)rep(k, P)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;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)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;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)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;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)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 _overloaddin(_1, _2, _3, _4, 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 din(...) _overloaddin(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__)

#define _overloaddind(_1, _2, _3, _4, 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(...) _overloaddind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)


#define _overloadout(_1, _2, _3, _4, name, ...) name
#define out1(a) cout<<a<<endl
#define out2(a, b) cout<<a<<" "<< b<<endl
#define out3(a, b, c) cout<<a<<" "<<b<<" "<<c<<endl
#define out4(a, b, c, d) cout<<a<<" "<<b<<" "<<c<<" "<<d<<endl
#define out(...) _overloadout((__VA_ARGS__),out4,out3,out2,out1)((__VA_ARGS__))

template<class T> void outl(vector<T> &a) { fora(v, a)cout << v << endl; }
template<class T> void na(vector<T> &a, int n) {a.resize(n);rep(i, n)cin >> a[i];}
#define dna(a, n) vi a(n); rep(_i,n) cin >> a[_i];
template<class T> void nao(vector<T> &a, int n) {    a.resize(n + 1);    a[0] = 0;    rep(i, n)cin >> a[i + 1];}
template<class T> void nad(vector<T> &a, int n) {    a.resize(n);    rep(i, n)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);    rep(i, n)cin >> a[i] >> b[i];}
#define dna2(a, b, n) vi a(n),b(n);rep(_i, n)cin >> a[_i] >> b[_i];
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;    rep(i, n)cin >> a[i + 1] >> b[i + 1];}
template<class T, class U> void na2d(vector<T> &a, vector<U> &b, int n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i], a[i]--, b[i]--;}
template<class T, class U, class V> void na3(vector<T> &a, vector<U> &b, vector<V> &c, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i];}
#define dna3(a, b, c, n) vi a(n),b(n),c(n);   rep(_i, n)cin >> a[_i] >> b[_i] >> c[_i];
template<class T, class U, class V> void na3d(vector<T> &a, vector<U> &b, vector<V> &c, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)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);  rep(_i, n){cin >> a[_i] >> b[_i] >> c[_i];a[_i]--,b[_i]--,c[_i]--;}
#define nt(a, h, w) resize(a,h,w);rep(_hi,h)rep(_wi,w) cin >> a[_hi][_wi];
#define ntd(a, h, w) rs(a,h,w);rep(_hi,h)rep(_wi,w) cin >> a[_hi][_wi], a[_hi][_wi]--;
#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');rep(_hi,1,h+1)rep(_wi,1,w+1) cin >> a[_hi][_wi];

//デバッグ
#define sp << " " <<

#define debugName(VariableName) # VariableName

#define _deb1(x)  debugName(x)<<" = "<<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 _overloadebug(_1, _2, _3, _4, _5, _6, _7, _8, _9, _10, name, ...) name

#ifdef _DEBUG
#define deb(...)  cerr<< _overloadebug(__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'

//よく使うクラス、構造体
class UnionFind {
public:
    vi par, rank, sizes;
    int n, trees;
    UnionFind(int n) : n(n), trees(n) {
        par.resize(n), rank.resize(n), sizes.resize(n);
        rep(i, n)par[i] = i, sizes[i] = 1;
    }
    int root(int x) {
        if (par[x] == x)return x;
        else return par[x] = root(par[x]);
    }
    int find(int x) { return root(x); }
    void unite(int x, int y) {
        x = root(x);
        y = root(y);
        if (x == y)return;
        if (rank[x] < rank[y])swap(x, y);
        trees--;
        par[y] = x;
        sizes[x] += sizes[y];
        if (rank[x] == rank[y])rank[x]++;
    }
    bool same(int x, int y) { return root(x) == root(y); }
    int size(int x) { return sizes[root(x)]; }
    //順不同 umapなので
    vec<vi> sets() {
        vec<vi> res(trees);
        umap<int, vi> map;
        rep(i, n) map[root(i)].push_back(i);
        int i = 0;
        for (auto &&p:map) {
            int r = p.fi;
            res[i].push_back(r);
            for (auto &&v:p.se) {
                if (r == v)continue;
                res[i].push_back(v);
            }
            i++;
        }
        return res;
    }
};

using bint =__int128;
using u32 = unsigned int;
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";rep(i, 30)a += a;rep(i, 1 << 17)cout << a << endl;cout << "OLE 出力長制限超過" << endl;exit(0);
}
void re() { assert(0 == 1); }
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) {    vi v(n);    rep(i, n)v[i] = rand(min, max);    return v;}
//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);        rep(i, n) {            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);    rep(i, n) {        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);    rep(i, 1, sz(a)) {        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);    rep(i, 1, sz(a)) {        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 {        rep(i, sz(a)) {            if (a[i] < b[i]) {                return b;            } else if (a[i] > b[i])return a;        }    }    return a;}
//strを整数として比較
string smin(str &a, str b) {    if (sz(a) < sz(b)) {        return a;    } else if (sz(a) > sz(b)) {        return b;    } else {        rep(i, sz(a)) {            if (a[i] < b[i]) {                return a;            } else if (a[i] > b[i])return b;        }    }    return a;}
template<typename V, typename T> int find(vector<V> &a,const T key) {rep(i, sz(a))if (a[i] == key)return i;return -1;}
template<typename V, typename T> P find(vector<vector<V >> &a,const T key) {rep(i, sz(a))rep(j, sz(a[0]))if (a[i][j] == key)return mp(i, j);return mp(-1, -1);}
template<typename V, typename U> T find(vector<vector<vector<V >>> &a,const U key) {rep(i, sz(a))rep(j, sz(a[0]))rep(k, sz(a[0][0]))if (a[i][j][k] == key)return mt(i, j, k);return mt(-1, -1, -1);}

template<typename T, typename _Pr> int count_if(const T &v, _Pr pred) { return pred(v); }
template<typename T, typename _Pr> int count_if(const vector<T> &vec, _Pr pred) { int ret=0;fora(v,vec)ret+=count_if(v,pred) ;return ret;}

template<typename V, typename T> int count2(V &a, const T k) { return a == k; }
template<typename V, typename T> int count2(vector<V> &a, const T k) {int ret = 0;fora(v, a)ret +=count(v, k);return ret;}
template<typename V, typename T> int count(vector<V> &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;}
template<typename V> int count_od(vector<V> &a) {return count_if(a,[](int v){return v&1 ;});}
template<typename V> int count_ev(vector<V> &a) {return count_if(a,[](int v){return !(v&1) ;});}
//algorythm


inline ll rev(ll a) {ll 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 T> vector<T> revv(vector<T> &a) {vector<T> b = a;reverse(all(b));return b;}
void inline rev(string &a) {reverse(all(a));}
constexpr int 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(ll 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 ret = 0;for (; v; v /= 10)ret += v % 10;return ret;}

struct sint {
    int v;
    sint(int v) : v(v) {}
    operator int() { return v; }
    int operator[](int i) { return (v / p10[i]) % 10; }
    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 top(int i, int k) {
        int len = keta(v);
        return set(len - 1 - i, k);
    }
    int set(int i, int k) {
        if (i < 0)return top(abs(i) - 1, k);
        return v += p10[i] * (k - (v / p10[i]) % 10);
    }
    int inc(int i, int k = 1) {
        return v += p10[i] * k;
    }
    int inct(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 dect(int i, int k = 1) {
        return v -= p10[keta(v) - i - 1] * k;
    }
#define op(t, o)template<class T>inline 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> inline int operator<<=(T r) { return v *= p10[r]; }
    template<class T> inline int operator<<(T r) { return v * p10[r]; }
    template<class T> inline int operator>>=(T r) { return v /= p10[r]; }
    template<class T> inline int operator>>(T r) { return v / p10[r]; }
};
int mask10(int v) { return p10[v] - 1; }
//変換系
template<class T> auto keys(T a) {vector<decltype((a.begin())->fi)> res;for (auto &&k :a)res.pb(k.fi);return res;}
template<class T> auto values(T a) {vector<decltype((a.begin())->se)> res;for (auto &&k :a)res.pb(k.se);return res;}

template<class T, class U> inline bool chma(T &a, const U &b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}
template<class U> inline bool chma(const U &b) { return chma(ma, b); }
template<class T, class U> inline bool chmi(T &a, const U &b) {
    if (b < a) {
        a = b;
        return true;
    }
    return false;
}
template<class U> inline bool chmi(const U &b) { return chmi(mi, b); }
template<class T> inline T min(T a, signed b) { return a < b ? a : b; }
template<class T> inline T max(T a, signed b) { return a < b ? b : a; }
template<class T> inline T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }
template<class T> inline T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }
template<class T> inline T min(vector<T> a) { return *min_element(all(a)); }
template<class T> inline T mini(vector<T> a) { return min_element(all(a)) - a.begin(); }
template<class T> inline T min(vector<T> a, int n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T> inline T min(vector<T> a, int s, int n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<class T> inline T max(vector<T> a) { return *max_element(all(a)); }
template<class T> inline T maxi(vector<T> a) { return max_element(all(a)) - a.begin(); }
template<class T> inline T max(vector<T> a, int n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T> inline 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> inline A max(A (&a)[N]) {    A res = a[0];    rep(i, N)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O> inline 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> inline 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> inline 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> inline 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> inline 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> inline A min(A (&a)[N]) {    A res = a[0];    rep(i, N)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O> inline 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> inline 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> inline 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> inline 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> inline 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) {T ret = 0;rep(i, sz(v))ret += v[i];return ret;}
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... V> inline auto sum(vector<T> &v, U head, V... tail) {
    auto ret = sum(v[0], tail...);
    rep(i, 1, min(sz(v), head))ret += sum(v[i], tail...);
    return ret;
}
void clear(PQ &q) { while (q.size())q.pop(); }
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> inline 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> inline vi imo(vector<T> &a, U
k) {
    vector<T> ret = a;
    rep(i, sz(ret) - 1)ret[i + 1] = ret[i] + (a[i] == k);
    return
            ret;
}
template<class T> inline 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> inline 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) {}
    inline T operator()(int l, int r) {
        assert(l <= r);
        return rui[r] - rui[l];
    }
    inline T operator[](int i) {
        return rui[i];
    }
};

template<class T> struct rruic {
    const T *rrui;
    rruic(T *ru) : rrui(ru) {}
    //n-1から-1へ
    inline T operator()(int l, int r) {
        assert(l >= r);
        return rrui[r] - rrui[l];
    }
    inline T operator[](int i) { return rrui[i]; }
};
template<class T> inline 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> inline ruiC<T> ruic(vector<T> &a) {
    vector<T> ret = ruiv(a);
    return ruiC<T>(ret);
}
//kと同じものの数
template<class T, class U> inline 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> inline ruiC<int> ruic(T &a, U k) {
    vi ret = ruiv(a, k);
    return ruiC<int>(ret);
}

//xor
template<class T> inline 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> inline 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> inline vector<T> ruimi(vector<T> &a) {
    int 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) {
    int 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) {
    int 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 T> void dec(vector<T> &a) { for (auto &u :a)dec(u, 1); }
template<class T, class U> void minu(T &a, U v = 1) { a -= v; }
template<class T, class U> void minu(vector<T> &a, U v = 1) { for (auto &u:a)dec(u, v); }
template<class T> void minu(vector<T> &a) { for (auto &u :a)dec(u, 1); }
template<class U> void minu(string &a, U v = 1) { for (auto &u :a)dec(u, v); }
void minu(string &a) { for (auto &u :a)dec(u, 1); }
inline bool inside(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; }
inline bool inside(int l, int v, int r) { return l <= v && v < r; }
#define ins inside
ll u(ll 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()
ll goldd(ll left, ll right, function<ll(ll)> calc) {    double GRATIO = 1.6180339887498948482045868343656;    ll lm = left + (ll) ((right - left) / (GRATIO + 1.0));    ll rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));    ll fl = calc(lm);    ll fr = calc(rm);    while (right - left > 10) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + (ll) ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    ll minScore = MAX(ll);    ll resIndex = left;    for (ll i = left; i < right + 1; i++) {        ll score = calc(i);        if (minScore > score) {            minScore = score;            resIndex = i;        }    }    return resIndex;}
ll goldt(ll left, ll right, function<ll(ll)> calc) {    double GRATIO = 1.6180339887498948482045868343656;    ll lm = left + (ll) ((right - left) / (GRATIO + 1.0));    ll rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));    ll fl = calc(lm);    ll fr = calc(rm);    while (right - left > 10) {        if (fl > fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + (ll) ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    if (left > right) {        ll l = left;        left = right;        right = l;    }    ll maxScore = MIN(ll);    ll resIndex = left;    for (ll i = left; i < right + 1; i++) {        ll score = calc(i);        if (maxScore < score) {            maxScore = score;            resIndex = i;        }    }    return resIndex;}
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(ll m, int keta) {return (m >> keta) & 1;}
int bget(ll m, int keta, int sinsuu) {m /= (ll) pow(sinsuu, keta);return m % sinsuu;}
ll bit(int n) { return (1LL << (n)); }
ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); }
int mask(int n) { return (1ll << n) - 1; }
#define bcou __builtin_popcountll
//最下位ビット
int lbit(int n) {return n & -n;}
//最上位ビット
int hbit(int n) {n |= (n >> 1);n |= (n >> 2);n |= (n >> 4);n |= (n >> 8);n |= (n >> 16);n |= (n >> 32);return n - (n >> 1);}
int hbitk(int n) {int k = 0;rer(i, 5) {int a = k + (1ll << i);int b = 1ll << a;if (b <= n)k += 1ll << i;}return k;}
//初期化は0を渡す
ll nextComb(ll &mask, int n, int r) {
    if (!mask)return mask = (1LL << r) - 1;
    ll x = mask & -mask; //最下位の1
    ll y = mask + x; //連続した下の1を繰り上がらせる
    ll 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
vl bitCombList(int n, int r) {
    vl res;
    int m = 0;
    while (nextComb(m, n, r)) {
        res.pb(m);
    }
    return res;
}
//大文字小文字を区別する
int altoiaZ(char c) {
    if ('A' <= c && c <= 'Z')return c - 'A';
    return c - 'a' + 26;
}
char itoalaZ(int i) {
    if (i < 26)return 'A' + i;
    return 'a' + i - 26;
}
//aもAも0を返す 基本小文字
int altoi(char c) {
    if ('A' <= c && c <= 'Z')return c - 'A';
    return c - 'a';
}
char itoal(int i) {
    return 'a' + i;
}
int ctoi(char c) { return c - '0'; }
char itoc(int i) { return i + '0'; }
int vtoi(vi &v) {
    int 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(int i) {
    vi res;
    while (i) {
        res.pb(i % 10);
        i /= 10;
    }
    rev(res);
    return res;
}
vi stov(string &a) {
    int n = sz(a);
    vi ret(n);
    rep(i, n) {
        ret[i] = a[i] - '0';
    }
    return ret;
}
vector<vector<int>> ctoi(vector<vector<char>> s, char c) {
    int n = sz(s), m = sz(s[0]);
    vector<vector<int>> res(n, vector<int>(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] := i番として圧縮されたものを返す
vi compress(vi &a) {
    vi b;
    int len = a.size();
    for (int i = 0; i < len; ++i) {
        b.push_back(a[i]);
    }
    sort(b);
    unique(b);
    for (int i = 0; i < len; ++i) {
        a[i] = lower_bound(all(b), a[i]) - b.begin();
    }
    int blen = sz(b);
    vi ret(blen);
    rep(i, blen) {
        ret[i] = b[i];
    }
    return ret;
}
vi compress(vi &a, umap<int, int> &map) {
    vi b;
    int len = a.size();
    for (int i = 0; i < len; ++i) {
        b.push_back(a[i]);
    }
    sort(b);
    unique(b);
    for (int i = 0; i < len; ++i) {
        int v = a[i];
        a[i] = lower_bound(all(b), a[i]) - b.begin();
        map[v] = a[i];
    }
    int blen = sz(b);
    vi ret(blen);
    rep(i, blen) {
        ret[i] = b[i];
    }
    return ret;
}

vi compress(vi &a, vi &r) {
    vi b;
    int len = a.size();
    fora(v, a)b.pb(v);
    fora(v, r)b.pb(v);
    sort(b);
    unique(b);
    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();
    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();
    int 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;
    int len = a.size();
    fora(v, a)b.pb(v);
    fora(v, r)b.pb(v);
    fora(v, s)b.pb(v);
    sort(b);
    unique(b);
    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();
    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();
    for (int i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();
    int blen = sz(b);
    vi ret(blen);
    rep(i, blen) {
        ret[i] = b[i];
    }
    return ret;
}
vi compress(_vvi &a) {
    vi b;
    fora(vv, a)fora(v, vv)b.pb(v);
    sort(b);
    unique(b);
    fora(vv, a)fora(v, vv)v = lower_bound(all(b), v) - b.begin();
    int 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.pb(v);
    sort(b);
    unique(b);
    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();
    int blen = sz(b);
    vi ret(blen);
    rep(i, blen) {
        ret[i] = b[i];
    }
    return ret;
}
void compress(int a[], int len) {
    vi b;
    for (int i = 0; i < len; ++i) {
        b.push_back(a[i]);
    }
    sort(b);
    unique(b);
    for (int 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 ans(a) cout<<a<<endl;continue;
#define poll(a) q.front();q.pop()
#define dpoll(a) q.front();q.pop_front()
#define pollLast(a) q.back();q.pop_back()
#define pollBack(a) q.back();q.pop_back()
template<class T> inline void fin(T s) { cout << s << endl, exit(0); }

//便利 数学 math
int mod(int a, int m) { return (a % m + m) % m; }
int pow(int a) { return a * a; };
ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }

ll comi(int n, int 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 comd(int n, int r) {    assert(n < 2020);    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];}
int gcd(int a, int b) {    while (b) a %= b, swap(a, b);    return abs(a);}
int gcd(vi b) {    ll res = b[0];    rep(i, 1, sz(b))res = gcd(b[i], res);    return res;}
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll lcm(vi a) {    int res = a[0];    rep(i, 1, sz(a))res = lcm(a[i], res);    return res;}
ll ceil(ll a, ll b) {    if (b == 0) {        debugline("ceil");        deb(a, b);        ole();        return -1;    } else if (a < 0) {        return 0;    } else {return (a + b - 1) / b;}}

//v * v >= aとなる最小のvを返す
ll sqrt(ll a) {if (a < 0) {debugline("sqrt");deb(a);ole();}ll res = (ll) std::sqrt(a);while (res * res < a)res++;return res;}
double log(double e, double x) { return log(x) / log(e); }
ll sig(ll t) { return (1 + t) * t / 2; }
ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }


//幾何 Pをcomplexとして扱う
template<class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }
dou atan2(pd a) { return atan2(a.se, a.fi); }
dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }
dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }
//bを中心とするabcのtheta aからcにかけて時計回り
dou angle(pd a, pd b, pd c) {    dou ax = a.fi - b.fi;    dou ay = a.se - b.se;    dou cx = c.fi - b.fi;    dou cy = c.se - b.se;    double ret = atan2(cy, cx) - atan2(ay, ax);    if (ret < 0) ret += 2 * PI;    return ret;}
dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }
dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }

//機能拡張
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, class U>priority_queue<T, vector<T>, greater<T> > &operator+=(priority_queue<T, vector<T>, greater<T> > &a, vector <U> &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, class U> set<T> &operator+=(set<T> &a, U v) {    a.insert(v);    return a;}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.pb(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) {fora(v, b)a += v;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<typename T> void erase(vector<T> &v, unsigned int i) {v.erase(v.begin()+ i);}
template<typename T> void erase(vector<T> &v, unsigned int s,unsigned int e) {v.erase(v.begin()+ s, v.begin()+ e);}
template<class T, class U> void erase(map<T, U> &m, int okl, int ngr) {m.erase(m.lower_bound(okl), m.lower_bound(ngr));}
template<class T> void erase(set<T> &m, int okl, int ngr) {m.erase(m.lower_bound(okl), m.lower_bound(ngr));}
template<typename T> void erasen(vector<T> &v, unsigned int s,unsigned int n) {v.erase(v.begin()+ s, v.begin()+ s + n);}
template<typename T, typename U> void insert(vector<T> &v, unsigned int 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 int 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;
    int 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) {
    string b = a + deli;
    int l = 0, r = 0, n = b.size(), dn = deli.size();
    vector<string> res;
    rep(i, n) {
        if (i + dn <= n && b.substr(i, i + dn) == deli) {
            r = i;
            if (l < r)res.push_back(b.substr(l, r - l));
            i += dn - 1;
            l = i + 1;
        }
    }
    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);}
//@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));
    };
} __initon;

//@formatter:on
//gra mint pr
int n, m, k, d, H, W, x, y, z, q;
int cou;
template<typename T> T minv(T a, T m);
template<typename T> T minv(T a);

template<typename T>
class Modular {
public:
    using Type = typename decay<decltype(T::value)>::type;

    constexpr Modular() : value() {}
    template<typename U>
    Modular(const U &x) {
        value = normalize(x);
    }

    template<typename U>
    static Type normalize(const U &x) {
        Type v;
        if (-mod() <= x && x < mod()) v = static_cast<Type>(x);
        else v = static_cast<Type>(x % mod());
        if (v < 0) v += mod();
        return v;
    }

    const Type &operator()() const { return value; }
    template<typename U>
    explicit operator U() const { return static_cast<U>(value); }
    constexpr static Type mod() { return T::value; }

    Modular &operator+=(const Modular &other) {
        if ((value += other.value) >= mod()) value -= mod();
        return *this;
    }
    Modular &operator-=(const Modular &other) {
        if ((value -= other.value) < 0) value += mod();
        return *this;
    }
    template<typename U> Modular &operator+=(const U &other) { return *this += Modular(other); }
    template<typename U> Modular &operator-=(const U &other) { return *this -= Modular(other); }
    Modular &operator++() { return *this += 1; }
    Modular &operator--() { return *this -= 1; }
    Modular operator++(signed) {
        Modular result(*this);
        *this += 1;
        return result;
    }
    Modular operator--(signed) {
        Modular result(*this);
        *this -= 1;
        return result;
    }
    Modular operator-() const { return Modular(-value); }

    template<typename U = T>
    typename enable_if<is_same<typename Modular<U>::Type, signed>::value, Modular>::type &operator*=(const Modular &rhs) {
#ifdef _WIN32
        uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);
        uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;
        asm(
        "divl %4; \n\t"
        : "=a" (d), "=d" (m)
        : "d" (xh), "a" (xl), "r" (mod())
        );
        value = m;
#else
        value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
#endif
        return *this;
    }
    template<typename U = T>
    typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type &operator*=(const Modular &rhs) {
        int64_t q = static_cast<int64_t>(static_cast<double>(value) * rhs.value / mod());
        value = normalize(value * rhs.value - q * mod());
        return *this;
    }
    template<typename U = T>
    typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type &operator*=(const Modular &rhs) {
        value = normalize(value * rhs.value);
        return *this;
    }
    Modular &operator/=(const Modular &other) { return *this *= Modular(minv(other.value)); }
    template<typename U> friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);
    template<typename U> friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);
    template<typename U> friend std::istream &operator>>(std::istream &stream, Modular<U> &number);
    operator int() { return normalize(value); }
private:
    Type value;
};

template<typename T> bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value == rhs.value; }
template<typename T, typename U> bool operator==(const Modular<T> &lhs, U rhs) { return lhs == Modular<T>(rhs); }
template<typename T, typename U> bool operator==(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) == rhs; }

template<typename T> bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) { return !(lhs == rhs); }
template<typename T, typename U> bool operator!=(const Modular<T> &lhs, U rhs) { return !(lhs == rhs); }
template<typename T, typename U> bool operator!=(U lhs, const Modular<T> &rhs) { return !(lhs == rhs); }

template<typename T> bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value < rhs.value; }

template<typename T> Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }
template<typename T, typename U> Modular<T> operator+(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) += rhs; }
template<typename T, typename U> Modular<T> operator+(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }

template<typename T> Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }
template<typename T, typename U> Modular<T> operator-(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) -= rhs; }
template<typename T, typename U> Modular<T> operator-(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }

template<typename T> Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }
template<typename T, typename U> Modular<T> operator*(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) *= rhs; }
template<typename T, typename U> Modular<T> operator*(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }

template<typename T> Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }
template<typename T, typename U> Modular<T> operator/(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) /= rhs; }
template<typename T, typename U> Modular<T> operator/(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }

constexpr signed MOD = 1e9 + 7;//MOD
using mint = Modular<std::integral_constant<decay<decltype(MOD)>::type, MOD>>;

constexpr int mint_len = 1400001;
vi fac, finv, inv;
vi p2;
void setmod() {
//    p2.resize(mint_len);p[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i] = p2[i - 1] * 2 % MOD;
    fac.resize(mint_len);
    finv.resize(mint_len);
    inv.resize(mint_len);
    inv[1] = fac[0] = finv[0] = 1;
    for (int i = 2; i < mint_len; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
    for (int i = 1; i < mint_len; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
}
#define smod setmod
mint com(int n, int r) {
    if (r < 0 || r > n) return 0;
    return mint(finv[r] * fac[n] % MOD * finv[n - r]);
}
mint pom(int n, int r) {
//    if (!sz(fac)) com(0, -1);
    if (r < 0 || r > n) return 0;
    return mint(fac[n] * finv[n - 1]);
}
mint npr(int n, int r) {
//    if (!sz(fac)) com(0, -1);
    if (r < 0 || r > n) return 0;
    return mint(fac[n] * finv[n - r]);
}
int nprin(int n, int r) {
//    if (!sz(fac)) com(0, -1);
    if (r < 0 || r > n) return 0;
    return fac[n] * finv[n - r] % MOD;
}

int icom(int n, int r) {
    const int NUM_ = 1400001;
    static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];
    if (fac[0] == 0) {
        inv[1] = fac[0] = finv[0] = 1;
        for (int i = 2; i <= NUM_; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;
        for (int i = 1; i <= NUM_; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
    }
    if (r < 0 || r > n) return 0;
    return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;
}
#define ncr com
#define ncri icom
mint nhr(int n, int r) { return com(n + r - 1, r); }
mint hom(int n, int r) { return com(n + r - 1, r); }
int nhri(int n, int r) { return icom(n + r - 1, r); }


template<typename T> T minv(T a, T m) {
    T u = 0, v = 1;
    while (a != 0) {
        T t = m / a;
        m -= t * a;
        swap(a, m);
        u -= t * v;
        swap(u, v);
    }
    assert(m == 1);
    return u;
}
template<typename T> T minv(T a) {
    if (a < mint_len)return inv[a];
    T u = 0, v = 1;
    T m = MOD;
    while (a != 0) {
        T t = m / a;
        m -= t * a;
        swap(a, m);
        u -= t * v;
        swap(u, v);
    }
    assert(m == 1);
    return u;
}

template<typename T, typename U> Modular<T> mpow(const Modular<T> &a, const U &b) {
    assert(b >= 0);
    int x = a(), res = 1;
    U p = b;
    while (p > 0) {
        if (p & 1) (res *= x) %= MOD;
        (x *= x) %= MOD;
        p >>= 1;
    }
    return res;
}
template<typename T, typename U, typename V> mint mpow(const T a, const U b, const V m = MOD) {
    assert(b >= 0);
    int x = a, res = 1;
    U p = b;
    while (p > 0) {
        if (p & 1) (res *= x) %= m;
        (x *= x) %= m;
        p >>= 1;
    }
    return res;
}
template<typename T, typename U> mint mpow(const T a, const U b) {
    assert(b >= 0);
    int x = a, res = 1;
    U p = b;
    while (p > 0) {
        if (p & 1) (res *= x) %= MOD;
        (x *= x) %= MOD;
        p >>= 1;
    }
    return res;
}
template<typename T, typename U, typename V> int mpowi(const T &a, const U &b, const V &m = MOD) {
    assert(b >= 0);
    int x = a, res = 1;
    U p = b;
    while (p > 0) {
        if (p & 1) (res *= x) %= m;
        (x *= x) %= m;
        p >>= 1;
    }
    return res;
}
template<typename T> string to_string(const Modular<T> &number) {
    return to_string(number());
}

template<typename T> std::ostream &operator<<(std::ostream &stream, const Modular<T> &number) {
    return stream << number();
}

template<typename T> std::istream &operator>>(std::istream &stream, Modular<T> &number) {
    typename common_type<typename Modular<T>::Type, int64_t>::type x;
    stream >> x;
    number.value = Modular<T>::normalize(x);
    return stream;
}

using PM = pair<mint, mint>;
using vm = vector<mint>;
using mapm = map<int, mint>;
using umapm = umap<int, mint>;
#define _overloadvvm(_1, _2, _3, _4, name, ...) name
#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)

#define vvm1(a) vec<vm> a
#define vvm2(a, b) vec<vm> a(b)
#define vvm3(a, b, c) vec<vm> a(b,vm(c))
#define vvm4(a, b, c, d) vec<vm> a(b,vm(c,d))
#define vvm(...) _overloadvvm(__VA_ARGS__,vvm4,vvm3,vvm2 ,vvm1)(__VA_ARGS__)

template<class T> struct edge {
    int f, t;
    T c;
    int id;
    int type;
    edge(int f, int t, T c = 1, int id = -1, int ty = -1) : f(f), t(t), c(c), id(id), type(ty) {}
    bool operator<(const edge &b) const { return c < b.c; }
    bool operator>(const edge &b) const { return c > b.c; }
};
template<class T> ostream &operator<<(ostream &os, edge<T> &e) {
    os << e.f << " " << e.t << " " << e.c;
    return os;
}
template<typename T> class graph {
protected:
    vector<bool> _used;
public :
    vector<vector<edge<T>>> g;
    vector<edge<T>> edges;
    int n;
    graph(int n) : n(n) { g.resize(n), _used.resize(n); }
    void clear() { g.clear(), edges.clear(); }
    void resize(int n) {
        this->n = n;
        g.resize(n);
        _used.resize(n);
    }
    int size() { return g.size(); }
    vector<edge<T> > &operator[](int i) { return g[i]; }
    virtual void add(int f, int t, T c, int id, int ty) = 0;
    virtual bool used(edge<T> &e) = 0;
    virtual bool used(int id) = 0;
    virtual void del(edge<T> &e) = 0;
    virtual void del(int id) = 0;
    virtual void set_edges() = 0;
};
template<typename T =ll> class digraph : public graph<T> {
public:
    using graph<T>::g;
    using graph<T>::n;
    using graph<T>::edges;
    using graph<T>::_used;
    int eid = 0;

    digraph(int n) : graph<T>(n) {}
    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {
            debugline("digraph add");
            deb(f, t, c, id, ty);
            ole();
        }
        if (id == -1)id = eid++;
        g[f].emplace_back(f, t, c, id, ty);
        edges.emplace_back(f, t, c, id, ty);
    }
    bool used(edge<T> &e) { return _used[e.id]; }
    bool used(int id) { return _used[id]; }
    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }
    void del(int id) { _used[id] = _used[id ^ 1] = 1; }
    void set_edges() {
        if (sz(edges))return;
        rep(i, n)fora(e, g[i])edges.pb(e);
    }
};
template<class T=int> class undigraph : public graph<T> {
public:
    using graph<T>::g;
    using graph<T>::n;
    using graph<T>::edges;
    using graph<T>::_used;
    int eid = 0;
    undigraph(int n) : graph<T>(n) {
    }
    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {
            debugline("undigraph add");
            deb(f, t, c, id, ty);
            ole();
        }
        if (id == -1)id = eid, eid += 2;
        g[f].emplace_back(f, t, c, id, ty);
        g[t].emplace_back(t, f, c, id + 1, ty);
        edges.emplace_back(f, t, c, id, ty);
        edges.emplace_back(t, f, c, id + 1, ty);
    }
    void add(edge<T> &e) {
        int f = e.f, t = e.t, ty = e.type;
        T c = e.c;
        add(f, t, c, ty);
    }
    bool used(edge<T> &e) { return _used[e.id]; }
    bool used(int id) { return _used[id]; }
    void del(edge<T> &e) { _used[e.id] = _used[e.id ^ 1] = 1; }
    void del(int id) { _used[id] = _used[id ^ 1] = 1; }
    void set_edges() {
        if (sz(edges))return;
        rep(i, n)fora(e, g[i])edges.pb(e);
    }
};
template<typename T> struct radixheap {
    vector<pair<u64, T> > v[65];
    u64 size, last;
    radixheap() : size(0), last(0) {}
    bool empty() const { return size == 0; }
    inline int getbit(int a) { return a ? 64 - __builtin_clzll(a) : 0; }
    void push(u64 key, const T &value) {
        ++size;
        v[getbit(key ^ last)].emplace_back(key, value);
    }
    pair<u64, T> pop() {
        if (v[0].empty()) {
            int idx = 1;
            while (v[idx].empty()) ++idx;
            last = min_element(begin(v[idx]), end(v[idx]))->first;
            for (auto &p : v[idx]) v[getbit(p.first ^ last)].emplace_back(p);
            v[idx].clear();
        }
        --size;
        auto ret = v[0].back();
        v[0].pop_back();
        return ret;
    }
};
vi dijkstra2(const graph<int> &g, int s, int cant_arrive = -1) {
    if (!(0 <= s && s < g.n)) {
        debugline("dijkstra");
        deb(s, g.n);
        ole();
    }
    int initValue = MAX(int);
    vi dis(g.n, initValue);
    radixheap<int> q;
    dis[s] = 0;
    q.push(0, s);
    while (!q.empty()) {
        int nowc, i;
        tie(nowc, i) = q.pop();
        if (dis[i] != nowc)continue;
        for (auto &&e  : g.g[i]) {
            int to = e.t;
            int c = nowc + e.c;
            if (dis[to] > c) {
                dis[to] = c;
                q.push(dis[to], to);
            }
        }
    }
    //基本、たどり着かないなら-1
    if (cant_arrive == -1)for (auto &&d :dis) if (d == initValue)d = -1;
    return dis;
}

template<class T> vector<T> dijkstra(const graph<T> &g, int s, int cant_arrive = -1) {
    if (!(0 <= s && s < g.n)) {
        debugline("dijkstra");
        deb(s, g.n);
        ole();
    }
    T initValue = MAX(T);
    vector<T> dis(g.n, initValue);
    priority_queue<pair<T, int>, vector<pair<T, int>>, greater<pair<T, int>>> q;
    dis[s] = 0;
    q.emplace(0, s);
    while (q.size()) {
        T nowc = q.top().fi;
        int i = q.top().se;
        q.pop();
        if (dis[i] != nowc)continue;
        for (auto &&e  : g.g[i]) {
            int to = e.t;
            T c = nowc + e.c;
            if (dis[to] > c) {
                dis[to] = c;
                q.emplace(dis[to], to);
            }
        }
    }
    //基本、たどり着かないなら-1
    if (cant_arrive == -1)for (auto &&d :dis) if (d == initValue)d = -1;
    return dis;
}
template<class T> vector<vector<T>> warshall(const graph<T> &g, int cant_arrive = -1) {
    int n = g.n;
    vector<vector<T> > dis(n, vector<T>(n, linf));
    rep(i, n)fora(e, g.g[i])chmi(dis[e.f][e.t], e.c);
    rep(i, n)dis[i][i] = 0;
    rep(k, n)rep(i, n)rep(j, n)chmi(dis[i][j], dis[i][k] + dis[k][j]);
    //基本、たどり着かないなら-1
    if (cant_arrive == -1)rep(i, n)rep(j, n) if (dis[i][j] == linf)dis[i][j] = -1;
    return dis;
}

template<class T=int> class tree : public undigraph<T> {
public:
    using undigraph<T>::g;
    using undigraph<T>::n;
    using undigraph<T>::edges;
    using undigraph<T>::_used;
    vi disv;
private:
    bool never = 1;
    int root = -1;
    vi dep;//到着した順の深さ
    vi vis;//到着した順
    vi first;//初めてiに到着した時
    vector<int> bestt;//タイミングを持つ
    int sn = 0;//seg木の底辺の長さ euler tourの長さ
    void built() {
        never = 0;
        n = g.size();
        //euler tour
        disv.resize(n);
        first.resize(n);
        dep.resize(n * 2);
        vis.resize(n * 2);
        function<void(int, int, int, int &)> dfs = [&](int i, int p, int d, int &k) {
            first[i] = k;
            vis[k] = i;
            dep[k++] = d;
            forg(gi, g[i]) {
                if (t == p)continue;
                disv[t] = disv[i] + c;
                dfs(t, i, d + 1, k);
                vis[k] = i;
                dep[k++] = d;
            }
        };
        int k = 0;
        dfs(root, -1, 0, k);
        //rmq
        sn = 1;
        while (sn < n * 2)sn <<= 1;

        bestt.resize((sn << 1) - 1);
        rep(i, k)bestt[i + sn - 1] = i;//時間を持つ
        rer(i, sn - 2) {
            int lt = (i << 1) + 1, rt = (i << 1) + 2;
            if (dep[bestt[lt]] < dep[bestt[rt]]) {
                bestt[i] = bestt[lt];
            } else {
                bestt[i] = bestt[rt];
            }
        }
    }
    //最小の深さを持つタイミングが知りたい
    inline int lca_rmq(int l, int r) {
        int midv = inf;
        int midi = -1;
        l += sn - 1;
        r += sn - 1;
        while (l < r) {
            //右なら、次に右上へ移動してしまうので
            if (!(l & 1)) {
                if (chmi(midv, dep[bestt[l]])) {
                    midi = bestt[l];
                }
            }
            //右なら、左を取らないと機会を逃す
            if (!(r & 1)) {
                if (chmi(midv, dep[bestt[r - 1]])) {
                    midi = bestt[r - 1];
                }
            }
            l >>= 1;
            r = (r - 1) >> 1;
        }
        return vis[midi];
    }

public:
    tree(int n, int root = 0) : undigraph<T>(n), root(root) {}
    bool leaf(int v) {
        return sz(g[v]) == 1 && v != root;
    }
    int lca(int u, int v) {
        if (never)built();
        if (first[u] > first[v])swap(u, v);
        return lca_rmq(first[u], first[v] + 1);
    }
    int dis(int u, int v) {
        if (never) built();
        if (first[u] > first[v])swap(u, v);
        int p = lca_rmq(first[u], first[v] + 1);
        return disv[u] + disv[v] - disv[p] * 2;
    }

};
//辺によりメモリを大量消費
// よってedgesを消している
//頂点10^6でメモリを190MB(制限の8割)使う

//軽量化のため、辺を持たないbig gridクラスがあってもいいかもしれない
//
template<class T=int> class grid_k6 : public undigraph<T> {
public:
    using undigraph<T>::g;
    using undigraph<T>::n;
    using undigraph<T>::edges;
    using undigraph<T>::_used;
    int H, W;
    int eid = 0;
    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {
            debugline("grid_k6 add");
            deb(f, t, c, id, ty);
            ole();
        }
        g[f].emplace_back(f, t, c, eid++, ty);
        g[t].emplace_back(t, f, c, eid++, ty);
    }
    int getid(int h, int w) {
        if (!inside(h, w, H, W))return -1;
        return W * h + w;
    }
    P get2(int id) {
        return mp(id / W, id % W);
    }
    P operator()(int id) {
        return get2(id);
    }
    int operator()(int h, int w) {
        return getid(h, w);
    }
    grid_k6(int H, int W) : H(H), W(W), undigraph<T>(H * W) {
        rep(h, H) {
            rep(w, W) {
                int f = getid(h, w);
                if (w + 1 < W) add(f, getid(h, w + 1));
                if (h + 1 < H)add(f, getid(h + 1, w));
            }
        }
    }
    grid_k6(_vvc ba, char wall = '#') : H(sz(ba)), W(sz(ba[0])), undigraph<T>(sz(ba) * sz(ba[0])) {
        rep(h, H) {
            rep(w, W) {
                if (ba[h][w] == wall)con;
                int f = getid(h, w);
                if (w + 1 < W && ba[h][w + 1] != wall) {
                    add(f, getid(h, w + 1));
                }
                if (h + 1 < H && ba[h + 1][w] != wall) {
                    add(f, getid(h + 1, w));
                }
            }
        }
    }
    void add(int fh, int fw, int th, int tw) {
        add(getid(fh, fw), getid(th, tw));
    }
    void set_edges() {
        rep(i, n)fora(e, g[i])edges.pb(e);
    }
};
//左上から右下に移動できる
template<class T=int> class digrid_k6 : public digraph<T> {
public:
    using digraph<T>::g;
    using digraph<T>::n;
    using digraph<T>::edges;
    using digraph<T>::_used;
    int H, W;
    int eid = 0;
    void add(int f, int t, T c = 1, int id = -1, int ty = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {
            debugline("digrid_k6 add");
            deb(f, t, c, id, ty);
            ole();
        }
        g[f].emplace_back(f, t, c, eid++, ty);
    }
    int getid(int h, int w) {
        if (!inside(h, w, H, W))return -1;
        return W * h + w;
    }
    P get2(int id) {
        return mp(id / W, id % W);
    }
    P operator()(int id) {
        return get2(id);
    }
    int operator()(int h, int w) {
        return getid(h, w);
    }
    digrid_k6(int H, int W) : H(H), W(W), digraph<T>(H * W) {
        rep(h, H) {
            rep(w, W) {
                int f = getid(h, w);
                if (w + 1 < W) add(f, getid(h, w + 1));
                if (h + 1 < H)add(f, getid(h + 1, w));
            }
        }
    }
    digrid_k6(_vvc ba, char wall = '#') : H(sz(ba)), W(sz(ba[0])), digraph<T>(sz(ba) * sz(ba[0])) {
        rep(h, H) {
            rep(w, W) {
                if (ba[h][w] == wall)con;
                int f = getid(h, w);
                if (w + 1 < W && ba[h][w + 1] != wall) {
                    add(f, getid(h, w + 1));
                }
                if (h + 1 < H && ba[h + 1][w] != wall) {
                    add(f, getid(h + 1, w));
                }
            }
        }
    }
    void add(int fh, int fw, int th, int tw) {
        add(getid(fh, fw), getid(th, tw));
    }
    void set_edges() {
        rep(i, n)fora(e, g[i])edges.pb(e);
    }
};
template<class T> bool nibu(const graph<T> &g) {
    int size = 0;
    rep(i, g.n)size += sz(g.g[i]);
    if (size == 0)return true;

    UnionFind uf(g.n * 2);
    rep(i, g.n)fora(e, g.g[i])uf.unite(e.f, e.t + g.n), uf.unite(e.f + g.n, e.t);
    rep(i, g.n)if (uf.same(i, i + g.n))return 0;
    return 1;
}
//二部グラフを色分けした際の頂点数を返す
template<class T> vp nibug(graph<T> &g) {
    vp cg;
    if (!nibu(g)) {
        debugline("nibu");
        ole();
    }
    int _n = g.size();
    vb _was(_n);
    queue<P> q;
    rep(i, _n) {
        if (_was[i])continue;
        q.push(mp(i, 1));
        _was[i] = 1;
        int red = 0;
        int coun = 0;
        while (q.size()) {
            int now = q.front().fi;
            int col = q.front().se;
            red += col;
            coun++;
            q.pop();
            forg(gi, g[now]) {
                if (_was[t])continue;
                q.push(mp(t, col ^ 1));
                _was[t] = 1;
            }
        }
        cg.push_back(mp(red, coun - red));
    }
    return cg;

}
template<class T> ostream &operator<<(ostream &os, graph<T> &g) {
    rep(i, sz(g)) {
        forg(gi, g[i]) {
            os << f << " " << t << " " << c << endl;
        }
    }
    return os;
}
//閉路がなければtrue
bool topo(vi &res, digraph<int> &g) {
    int n = g.g.size();
    vi nyu(n);
    rep(i, n)for (auto &&e :g[i])nyu[e.t]++;
    queue<int> st;
    rep(i, n)if (nyu[i] == 0)st.push(i);
    while (st.size()) {
        int v = st.front();
        st.pop();
        res.pb(v);
        fora(e, g[v]) if (--nyu[e.t] == 0)st.push(e.t);
    }
    return res.size() == n;
}
//辞書順最小トポロジカルソート
bool topos(vi &res, digraph<int> &g) {
    int n = g.g.size();
    vi nyu(n);
    rep(i, n)for (auto &&e :g[i])nyu[e.t]++;
    //小さい順
    priority_queue<int, vector<int>, greater<int> > q;
    rep(i, n)if (nyu[i] == 0)q.push(i);
    while (q.size()) {
        int i = q.top();
        q.pop();
        res.pb(i);
        fora(e, g[i])if (--nyu[e.t] == 0)q.push(e.t);
    }
    return res.size() == n;
}


vi t, a, b, c;
//vvi (s, 0, 0);
vvc (ba, 0, 0);
vp p;
str s;
undigraph<> g(2 * k5);
vi v(2 * k5);
void ds(int p, int i) {
    v[i] = c[k];
    k++;
    fort(gi, g[i]) {
            ds(i, t);
        }

}

void solve() {
    cin >> n;
    rep(i, n - 1) {
        int f, t;
        cin >> f >> t;
        --f, --t;
        g.add(f, t);
    }
    na(c, n);
    rsort(c);
    cout << sum(c) - c[0] << endl;
    ds(-1, 0);
    rep(i,n){
        cout <<v[i]<<" ";
    }
    cout << "" << endl;
}

int my(int n, vi &a) {
    return 0;
}
int sister(int n, vi &a) {
    int 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);
        int myres = my(arg);
        int res = sister(arg);
        ok = myres == res;
        if (!ok) {
            out(arg);
            cerr << "正解 : " << res << endl;
            cerr << "自分  : " << 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;
};


Measured memory_bytes: 8980