//#undef _DEBUG
//#pragma GCC optimize ("-O3")
#include <bits/stdc++.h>
using namespace std;
//@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;
//衝突対策
#define ws ___ws
struct T {
    int f, s, t;
    T() { f = -1, s = -1, t = -1; }
    T(int f, int s, int t) : f(f), s(s), t(t) {}
    bool operator<(const T &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; 大きい順
    }
    bool operator>(const T &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 T &r) const {
        return f == r.f && s == r.s && t == r.t;
    }
    bool operator!=(const T &r) const {
        return f != r.f || s != r.s || t != r.t;
    }
    int operator[](int i) {
        assert(i < 3);
        return i == 0 ? f : i == 1 ? s : t;
    }
};
#define int long long
#define ll long long
#define double long double
#define pow powl
#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>;

#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 ==


//マクロ省略系 コンテナ


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

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

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

template<typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template<typename T, typename... Ts> auto make_v(size_t a, Ts... ts) { return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...)); }
#define vni(name, ...) auto name = make_v<ll>(__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 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>

#define PQ priority_queue<ll, vector<ll>, greater<ll> >
#define tos to_string
using mapi = map<int, 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 _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,)(__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 fora(a, b) for(auto&& a : b)

//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 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 rs resize
#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()


inline void sort(string &a) { sort(a.begin(), a.end()); }
template<class T> inline void sort(vector<T> &a) { sort(a.begin(), a.end()); };
template<class T> inline void sort(vector<T> &a, int len) { sort(a.begin(), a.begin() + len); };
template<class T, class F> inline void sort(vector<T> &a, F f) { sort(a.begin(), a.end(), [&](T l, T r) { return f(l) < f(r); }); };
enum ___pcomparator {
    fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd
};
inline void sort(vector<P> &a, ___pcomparator type) {
    switch (type) {
        case fisi:
            sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });
            break;
        case fisd:
            sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });
            break;
        case fdsi:
            sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });
            break;
        case fdsd:
            sort(all(a), [&](P l, P r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });
            break;
        case sifi:
            sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });
            break;
        case sifd:
            sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });
            break;
        case sdfi:
            sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });
            break;
        case sdfd:
            sort(all(a), [&](P l, P r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });
            break;
    }
};
inline void sort(vector<T> &a, ___pcomparator type) {
    switch (type) {
        case fisi:
            sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });
            break;
        case
            fisd:
            sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });
            break;
        case
            fdsi:
            sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });
            break;
        case
            fdsd:
            sort(all(a), [&](T l, T r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });
            break;
        case
            sifi:
            sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });
            break;
        case
            sifd:
            sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });
            break;
        case
            sdfi:
            sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });
            break;
        case
            sdfd:
            sort(all(a), [&](T l, T r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });
            break;
    }
};
template<class T> inline void rsort(vector<T> &a) { sort(a.begin(), a.end(), greater<T>()); };
template<class T> inline void rsort(vector<T> &a, int len) { sort(a.begin(), a.begin() + len, greater<T>()); };
template<class U, class F> inline void rsort(vector<U> &a, F f) { sort(a.begin(), a.end(), [&](U l, U r) { return f(l) > f(r); }); };
template<class U> inline void sortp(vector<U> &a, vector<U> &b) {
    vp c;
    int n = sz(a);
    assert(n == sz(b));
    rep(i, n)c.eb(a[i], b[i]);
    sort(c);
    rep(i, n) {
        a[i] = c[i].first;
        b[i] = c[i].second;;
    }
};
//F = T<T>
//例えばreturn p.fi + p.se;
template<class U, class F> inline void sortp(vector<U> &a, vector<U> &b, F f) {
    vp c;
    int n = sz(a);
    assert(n == sz(b));
    rep(i, n)c.eb(a[i], b[i]);
    sort(c, f);
    rep(i, n) {
        a[i] = c[i].first;
        b[i] = c[i].second;
    }
};
template<class U, class F> inline void sortp(vector<U> &a, vector<U> &b, char type) {
    vp c;
    int n = sz(a);
    assert(n == sz(b));
    rep(i, n)c.eb(a[i], b[i]);
    sort(c, type);
    rep(i, n) {
        a[i] = c[i].first;
        b[i] = c[i].second;
    }
};
template<class U> inline void rsortp(vector<U> &a, vector<U> &b) {
    vp c;
    int n = sz(a);
    assert(n == sz(b));
    rep(i, n)c.eb(a[i], b[i]);
    rsort(c);
    rep(i, n) {
        a[i] = c[i].first;
        b[i] = c[i].second;
    }
};
template<class U, class F> inline void rsortp(vector<U> &a, vector<U> &b, F f) {
    vp c;
    int n = sz(a);
    assert(n == sz(b));
    rep(i, n)c.eb(a[i], b[i]);
    rsort(c, f);
    rep(i, n) {
        a[i] = c[i].first;
        b[i] = c[i].second;
    }
};
template<class U> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c) {
    vt r;
    int n = sz(a);
    assert(n == sz(b));
    assert(n == sz(c));
    rep(i, n)r.eb(a[i], b[i], c[i]);
    sort(r);
    rep(i, n) {
        a[i] = r[i].f;
        b[i] = r[i].s;
        c[i] = r[i].t;
    }
};
template<class U, class F> inline void sortt(vector<U> &a, vector<U> &b, vector<U> &c, F f) {
    vt r;
    int n = sz(a);
    assert(n == sz(b));
    assert(n == sz(c));
    rep(i, n)r.eb(a[i], b[i], c[i]);
    sort(r, f);
    rep(i, n) {
        a[i] = r[i].f;
        b[i] = r[i].s;
        c[i] = r[i].t;
    }
};
template<class U, class F> inline void rsortt(vector<U> &a, vector<U> &b, vector<U> &c, F f) {
    vt r;
    int n = sz(a);
    assert(n == sz(b));
    assert(n == sz(c));
    rep(i, n)r.eb(a[i], b[i], c[i]);
    rsort(r, f);
    rep(i, n) {
        a[i] = r[i].f;
        b[i] = r[i].s;
        c[i] = r[i].t;
    }
};
template<class T> inline void sort2(vector<vector<T>> &a) { for (int i = 0, n = a.size(); i < n; i++)sort(a[i]); }
template<class T> inline 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);
}

//@汎用便利関数 入力
template<typename T = int> T _in() {
    T x;
    cin >> x;
    return (x);
}
#define _overloadin(_1, _2, _3, _4, name, ...) name
#define in0() _in()
#define in1(a) cin>>a
#define in2(a, b) cin>>a>>b
#define in3(a, b, c) cin>>a>>b>>c
#define in4(a, b, c, d) cin>>a>>b>>c>>d
#define in(...) _overloadin(__VA_ARGS__,in4,in3,in2 ,in1,in0)(__VA_ARGS__)

#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(...) _overloadin(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__)

#define _overloaddind(_1, _2, _3, _4, name, ...) name
#define din1d(a) int a;cin>>a;a--
#define din2d(a, b) int a,b;cin>>a>>b;a--,b--
#define din3d(a, b, c) int a,b,c;cin>>a>>b>>c;a--,b--,c--
#define din4d(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d;;a--,b--,c--,d--
#define dind(...) _overloaddind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)


string sin() { return _in<string>(); }
ll lin() { return _in<ll>(); }
#define na(a, n) a.resize(n); rep(nai,n) cin >> a[nai];
#define nao(a, n) a.resize(n+1); rep(i,n) cin >> a[i+1];
#define nad(a, n) a.resize(n); rep(i,n){ cin >> a[i]; a[i]--;}
#define na2(a, b, n) a.resize(n),b.resize(n);rep(i, n)cin >> a[i] >> b[i];
#define na2d(a, b, n) a.resize(n),b.resize(n);rep(i, n){cin >> a[i] >> b[i];a[i]--,b[i]--;}
#define na3(a, b, c, n) a.resize(n),b.resize(n),c.resize(n);   rep(i, n)cin >> a[i] >> b[i] >> c[i];
#define na3d(a, b, c, 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 nt(a, h, w) a.resize(h);rep(i,h)a[i].resize(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) fill(a,'#');rep(hi,1,h+1)rep(wi,1,w+1) cin >> a[hi][wi];

#ifdef _DEBUG
template<class T> void err2(T &&head) { cerr << head; }
template<class T, class... U> void err2(T &&head, U &&... tail) {
    cerr << head << " ";
    err2(tail...);
}
template<class T, class... U> void err(T &&head, U &&... tail) {
    cerr << head << " ";
    err2(tail...);
    cerr << "" << endl;
}
template<class T> void err(T &&head) { cerr << head << endl; }
void err() { cerr << "" << endl; }
template<class T> string out_m2(vector<T> &a, ll W = inf) {
    stringstream ss;
    if (W == inf)W = min(sz(a), 12ll);
    if (sz(a) == 0)return ss.str();
    rep(i, W) { ss << a[i] << " "; }
    return ss.str();
}
template<class T> string out_m2(vector<vector<T> > &a, ll H = inf, ll W = inf, int key = -1) {
    H = min({H, sz(a), 12ll});
    W = min({W, sz(a[0]), 12ll});
    stringstream ss;
    ss << endl;
    if (key == -1)ss << " *|"; else ss << " " << key << "|";
    rep(w, W)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    rep(w, W * 4 + 3)ss << "_";
    ss << "" << endl;
    rep(h, H) {
        ss << std::right << std::setw(2) << h << "|";
        rep(w, min(sz(a[h]), 12ll)) { if (abs(a[h][w]) == linf) ss << "   e" << ""; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    return ss.str();
}
/*@formatter:off*/
template<class T> string out_m2(vector<vector<vector<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {stringstream ss;    if (H == inf)H = 5;    H = min(H, sz(a));    rep(i, H) {        ss << endl;        ss << out_m2(a[i], W, U, i);    }       return ss.str();}
string out_m2(int a) {stringstream ss;ss << a;return ss.str();}
template<class T> string out_m2(T &a) {stringstream ss;ss << a;return ss.str();}

template<class T> string out_m(vector<T> &a, ll W = inf) {stringstream ss;    if (W == inf)W = min(sz(a), 12ll);   if(sz(a)==0)return ss.str();   rep(i, W) { ss << a[i] << " "; }    ss << "" << endl;    return ss.str();}
template<class T> string out_m(vector<vector<T> > &a, ll H = inf, ll W = inf, int key = -1) {
    H = min({H, sz(a), 12ll});
    W = min({W, sz(a[0]), 12ll});
    stringstream ss;
    ss << endl;
    if (key == -1)ss << " *|"; else ss << " " << key << "|";
    rep(w, W)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    rep(w, W * 4 + 3)ss << "_";
    ss << "" << endl;
    rep(h, H) {
        ss << std::right << std::setw(2) << h << "|";
        rep(w, min(sz(a[h]),12ll)) { if (abs(a[h][w]) == linf) ss << "   e" << ""; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    ss << endl;
    return ss.str();
}
/*@formatter:off*/
template<class T> string out_m(vector<vector<vector<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {stringstream ss;    if (H == inf)H = 5;    H = min(H, sz(a));    rep(i, H) {        ss << endl;        ss << out_m(a[i], W, U, i);    }    ss << endl;    return ss.str();}
string out_m(int a) {stringstream ss;ss << a << endl;return ss.str();}
template<class T> string out_m(T &a) {stringstream ss;ss << a << endl;return ss.str();}
template<class T> void outv(vector<T> &a, ll W=inf) {cout << out_m(a,W) << endl;}
template<class T> void outv(vector<vector<T> > &a, ll H = linf, ll W = linf,int key=-1) {    cout << out_m(a,H,W,key) << endl;}
template<class T> void outv(vector<vector<vector<T> > > &a, ll H = linf, ll W = linf,ll U = linf) {cout << out_m(a,H,W,U)<< endl;}
#else
template<class T> void outv(vector<T> &a, ll W = inf) {
    rep(i, min(W, sz(a))) { cout << a[i] << " "; }
    cout << "" << endl;
}
template<class T> void outv(vector<vector<T> > &a, ll H = linf, ll W = linf, int key = -1) { rep(i, min(H, sz(a))) { outv(a[i], W); }}
template<class T> void outv(vector<vector<vector<T> > > &a, ll H = linf, ll W = linf, ll U = linf) { ; }
#define err(...);
#endif

#define debugName(VariableName) # VariableName

#define deb1(x)  debugName(x)<<" = "<<out_m2(x)
#define deb2(x, ...) deb1(x) <<", "<< deb1(__VA_ARGS__)
#define deb3(x, ...) deb1(x) <<", "<< deb2(__VA_ARGS__)
#define deb4(x, ...) deb1(x) <<", "<< deb3(__VA_ARGS__)
#define deb5(x, ...) deb1(x) <<", "<< deb4(__VA_ARGS__)
#define deb6(x, ...) deb1(x) <<", "<< deb5(__VA_ARGS__)
#define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)
#define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)
#define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)
#define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)

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

#ifdef _DEBUG
#define deb(...)  cerr<< o_ebug(__VA_ARGS__,deb10,deb9,deb8,deb7,deb6,deb5,deb4,deb3,deb2,deb1)(__VA_ARGS__) <<endl
#else
#define deb(...) ;
#endif
#define debugline(x) cerr << x << " " << "(L:" << __LINE__ << ")" << '\n'
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 tle() { while (inf)cout << inf << endl; }
ll gcd(ll a, ll b) { return b ? gcd(b, a % b) : a; }
ll gcd(vi b) {
    ll res = b[0];
    for (auto &&v :b)res = gcd(v, res);
    return res;
}
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
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));
}
void rev(string &a) {
    reverse(all(a));
}
ll ceil(ll a, ll b) {
    if (b == 0) {
        debugline("ceil");
        deb(a, b);
        ole();
        return -1;
    } else return (a + b - 1) / b;
}
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; }


/*@formatter:off*/
inline bool inside(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; }
inline bool inside(int v, int l, 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()

void yn(bool a) {
    if (a)cout << "yes" << endl;
    else cout << "no" << endl;
}
void Yn(bool a) {
    if (a)cout << "Yes" << endl;
    else cout << "No" << endl;
}
void YN(bool a) {
    if (a)cout << "YES" << endl;
    else cout << "NO" << endl;
}
void fyn(bool a) {
    if (a)cout << "yes" << endl;
    else cout << "no" << endl;
    exit(0);
}
void fYn(bool a) {
    if (a)cout << "Yes" << endl;
    else cout << "No" << endl;
    exit(0);
}
void fYN(bool a) {
    if (a)cout << "YES" << endl;
    else cout << "NO" << endl;
    exit(0);
}
void Possible(bool a) {
    if (a)cout << "Possible" << endl;
    else cout << "Impossible" << endl;
    exit(0);
}
void POSSIBLE(bool a) {
    if (a)cout << "POSSIBLE" << endl;
    else cout << "IMPOSSIBLE" << endl;
    exit(0);
}
template<class T, class U> set<T> &operator+=(set<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> T sum(vector<T> &v, int s = 0, int t = inf) {
    T ret = 0;
    rep(i, s, min(sz(v), t))ret += v[i];
    return ret;
}
void mod(int &a, int m) { a = (a % m + m) % m; }
template<class F> inline int mgr(int ok, int ng, F f) {
#define _mgrbody int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid;
    if (ok < ng)while (ng - ok > 1) { _mgrbody } else while (ok - ng > 1) { _mgrbody }
    return ok;
}

template<class F> inline int mgr(int ok, int ng, int second, F f) {
#define _mgrbody2 int mid = (ok + ng) / 2; if (f(mid, second))ok = mid; else ng = mid;
    if (ok < ng) while (ng - ok > 1) { _mgrbody2 } else while (ok - ng > 1) { _mgrbody2 }
    return ok;
}
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, ll>> run_length(string &a) {    vector<pair<char, ll>> 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<typename T> ostream &operator<<(ostream &os, vector<T> &m) {
    for (auto &&v:m) os << v << " ";
    return os;
}
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


//

//imoは0-indexed
//ruiは1-indexed
template<class T> vector<T> imo(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];    return ret;}
//kと同じものの数
template<class T, class U> vi imo(vector<T> &a, U k) {vector<T> ret = a;rep(i, sz(ret))ret[i] = a[i] == k;rep(i, sz(ret) - 1)ret[i + 1] += ret[i];return ret;}
template<class T> vector<T> imox(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}
//漸化的に最小を持つ
template<class T> vector<T> imi(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}
template<class T> struct ruiC {    const vector<T> rui;    ruiC(vector<T> &ru) : rui(ru) {}    T operator()(ll l, ll r) {        if (l > r) {            cerr<<"ruic ";deb(l, r);assert(0);        }        return rui[r] - rui[l];    }    T operator[](ll i) { return rui[i]; }    T back() { return rui.back(); }    ll size() { return rui.size(); }};
template<class T> struct rruic {    const T *rrui;    rruic(T *ru) : rrui(ru) {}     T operator()(ll l, ll r) {        assert(l >= r);        return rrui[r] - rrui[l];    }     T operator[](ll i) { return rrui[i]; }};
template<class T>ostream &operator<<(ostream &os, ruiC<T> a) {fora(v,a.rui)os<<v<<" ";return os;}
template<class T> vector<T> ruiv(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i];    return ret;}
template<class T> ruiC<T> ruic(vector<T> &a) {    vector<T> ret = ruiv(a);    return ruiC<T>(ret);}
vector<ll> ruiv(string &a) {    if (sz(a) == 0)return vi(1);    ll dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;    vector<ll> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec;    return ret;}
ruiC<ll> ruic(string &a) {    vector<ll> ret = ruiv(a);    return ruiC<ll>(ret);}
//kと同じものの数
template<class T, class U> vi ruiv(T &a, U k) {    vi ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + (a[i] == k);    return ret;}
template<class T, class U> ruiC<ll> ruic(T &a, U k) {    vi ret = ruiv(a, k);    return ruiC<ll>(ret);}
//xor
template<class T> struct ruixC {    const vector<T> rui;    ruixC(vector<T> &ru) : rui(ru) {}    T operator()(ll l, ll r) {        if (l > r) {            cerr << "ruiXc ";            deb(l, r);            assert(0);        }        return rui[r] ^ rui[l];    }    T operator[](ll i) { return rui[i]; }    T back() { return rui.back(); }    ll size() { return rui.size(); }};
template<class T> vector<T> ruix(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] ^ a[i];    return ret;}
template<class T> ruixC<ll> ruixc(vector<T>  &a) {vi ret = ruix(a);return ruixC<ll>(ret);}
template<class T> vector<T> ruim(vector<T> &a) {    vector<T> res(a.size() + 1, 1);    rep(i, a.size())res[i + 1] = res[i] * a[i];    return res;}
//漸化的に最小を1indexで持つ
template<class T> vector<T> ruimi(vector<T> &a) {    ll 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) {    ll 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) {    ll 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> 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); }
#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );
int max(vi &a) {
    int res = a[0];
    fora(v, a) {
        res = max(res, v);
    }
    return res;
}
int min(vi &a) {
    int res = a[0];
    fora(v, a) {
        res = min(res, v);
    }
    return res;
}
template<typename T> class fixed_point        : T {public:    explicit constexpr fixed_point(T &&t) noexcept: T(std::forward<T>(t)) {}    template<typename... Args> constexpr decltype(auto) operator()(Args &&... args) const { return T::operator()(*this, std::forward<Args>(args)...); }};template<typename T> static inline constexpr decltype(auto) fix(T &&t) noexcept { return fixed_point<T>{std::forward<T>(t)}; }
#define lam(right) [&](auto& p){return p right;}
template<typename W, typename H> void resize(vector<W> &vec, const H head) { vec.resize(head); }template<typename W, typename H, typename ... T> void resize(vector<W> &vec, const H &head, const T ... tail) {vec.resize(head);for (auto &v: vec)resize(v, tail...);}
template<typename T, typename F> bool all_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool all_of2(vector<T> &v, F f) {    rep(i, sz(v)) { if (!all_of2(v[i], f))return false; }    return true;}
template<typename T, typename F> bool any_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool any_of2(vector<T> &v, F f) {    rep(i, sz(v)) { if (any_of2(v[i], f))return true; }    return false;}
template<typename T, typename F> bool none_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool none_of2(vector<T> &v, F f) {    rep(i, sz(v)) { if (none_of2(v[i], f))return false; }    return true;}
template<typename T, typename F> bool find_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll find_if2(vector<T> &v, F f) {    rep(i, sz(v)) { if (find_if2(v[i], f))return i; }    return sz(v);}
template<typename T, typename F> bool rfind_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll rfind_if2(vector<T> &v, F f) {    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }    return -1;}
template<class T> bool contains(string &s, const T &v) { return s.find(v) != string::npos; }
template<typename T> bool contains(vector<T> &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); }
template<typename T, typename F> bool contains_if2(vector<T> &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); }
template<typename T, typename F> ll count_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll count_if2(vector<T> &vec, F f) {    ll ret = 0;    fora(v, vec)ret += count_if2(v, f);    return ret;}
template<typename T, typename F> void for_each2(T &v, F f) { f(v); }
template<typename T, typename F> void for_each2(vector<T> &vec, F f) { fora(v, vec)for_each2(v, f); }
template<typename W> ll count_od(vector<W> &a) {return count_if2(a,[](ll v){return v&1 ;});}
template<typename W> ll count_ev(vector<W> &a) {return count_if2(a,[](ll v){return !(v&1) ;});}
//削除された要素の数を返す
template<typename T, typename F> int erase_if2(vector<T> &v, F f) {vector<T> nv;int cou = 0;rep(i, sz(v)) {if (f(v[i])) {cou++;} else {nv.push_back(v[i]);}}v = nv;return cou;}
template<typename T, typename F> int erase_if2(vector<vector<T>> &v, F f) {int cou = 0;rep(i, sz(v)) {cou += erase_if2(v[i], f);}return cou;}
#define all_of(a,right) all_of2(a,lam(right))
#define all_of_f(a,f) all_of2(a,f)
#define any_of(a,right) any_of2(a,lam(right))
#define any_of_f(a,f) any_of2(a,f)
#define none_of(a,right) none_of2(a,lam(right))
#define none_of_f(a,f) none_of2(a,f)
#define find_if(a,right) find_if2(a,lam(right))
#define find_if_f(a,f) find_if2(a,f)
#define rfind_if(a,right) rfind_if2(a,lam(right))
#define rfind_if_f(a,f) rfind_if2(a,f)
#define contains_if(a,right) contains_if2(a,lam(right))
#define contains_if_f(a,f) contains_if2(a,f)
#define count_if(a, right) count_if2(a,lam(right))
#define count_if_f(a, f) count_if2(a,f)
#define for_each(a, right) do{fora(v,a){v right;}}while(0)
#define for_each_f(a, f) do{fora(v,a){f(v);}}while(0)
#define erase_if(a,right) erase_if2(a,lam(right))
#define erase_if_f(a,f) erase_if2(a,f)

/*@formatter:on*/

int N, K, H, W;
vi A, B, C;
/*@formatter:off*/
#define forg(gi, ve) for (ll gi = 0,forglim = ve.size(), f, t, c; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)
#define fort(gi, ve) for (ll 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 fore(gi, ve) for (ll gi = 0,forglim = ve.size(), f, t, c; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)

template<class T> struct edge {    int f, t;    T c;    edge(int f, int t, T c = 1) : f(f), t(t), c(c) {}    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 {
public :
    vector<vector<edge<T>>> g;
    vector<edge<T>> edges;
    int n;
    explicit graph(int n) : n(n) { g.resize(n); }
    void clear() { g.clear(), edges.clear(); }
    void resize(int n) {this->n = n;g.resize(n);}
    int size() { return n; }
    vector<edge<T> > &operator[](int i) { return g[i]; }
    virtual void add(int f, int t, T c) = 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;
    explicit digraph(int n) : graph<T>(n) {}
    void add(int f, int t, T c = 1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {debugline("digraph add");deb(f, t, c);ole();}
        g[f].emplace_back(f, t, c);
        edges.emplace_back(f, t, c);//edgesを使わないなら消せる
    }
    void ing(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t;            cin >> f >> t;            f -= minus;            t -= minus;            add(f, t);        }    }
    void ingc(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t, c;            cin >> f >> t >> c;            f -= minus;            t -= minus;            add(f, t, c);        }    }
    void set_edges() override{        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
template<class T=int> class undigraph : public graph<T> {
public:
    using graph<T>::g;    using graph<T>::n;    using graph<T>::edges;
    explicit undigraph(int n) : graph<T>(n) {}
    // f < t
    void add(int f, int t, T c = 1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("undigraph add");            deb(f, t, c);            ole();        }
        g[f].emplace_back(f, t, c);
        g[t].emplace_back(t, f, c);
        edges.emplace_back(f, t, c);//edgesを使わないなら消せる
        edges.emplace_back(t, f, c);
    }
    void add(edge<T> &e) {        int f = e.f, t = e.t;        T c = e.c;        add(f, t, c);    }
    void ing(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t;            cin >> f >> t;            f -= minus;            t -= minus;            add(f, t);        }    }
    void ingc(int n, int m, int minus = 1) {        this->resize(n);        rep(i, m) {            int f, t, c;            cin >> f >> t >> c;            f -= minus;            t -= minus;            add(f, t, c);        }    }
    void set_edges () override{        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
template<class T> class MinOp { public: T operator()(T a, T b) { return min(a, b); }};
template<typename OpFunc> struct SparseTable {    OpFunc op;    signed size;    vector<signed> lg;    vector<vector<pair<signed, signed>>> table;    void init(const vector<pair<signed, signed>> &array, OpFunc opfunc) {        signed n = array.size();        op = opfunc;        lg.assign(n + 1, 0);        for (signed i = 1; i <= n; i++) { lg[i] = 31 - __builtin_clz(i); }        table.assign(lg[n] + 1, array);        for (signed i = 1; i <= lg[n]; i++) { for (signed j = 0; j < n; j++) { if (j + (1 << (i - 1)) < n) { table[i][j] = op(table[i - 1][j], table[i - 1][j + (1 << (i - 1))]); } else { table[i][j] = table[i - 1][j]; }}}    }    pair<signed, signed> query(signed l, signed r) {        assert(l < r);        return op(table[lg[r - l]][l], table[lg[r - l]][r - (1 << lg[r - l])]);    }};
struct PMORMQ {    vector<signed> a;    SparseTable<MinOp<pair<signed, signed> > > sparse_table;    vector<vector<vector<signed> > > lookup_table;    vector<signed> block_type;    signed block_size, n_block;    void init(const vector<signed> &array) {        a = array;        signed n = a.size();        block_size = std::max(1, (31 - __builtin_clz(n)) / 2);        while (n % block_size != 0) {            a.push_back(a.back() + 1);            n++;        }        n_block = n / block_size;        vector<pair<signed, signed> > b(n_block, make_pair(INT_MAX, INT_MAX));        for (signed i = 0; i < n; i++) { b[i / block_size] = min(b[i / block_size], make_pair(a[i], i)); }        sparse_table.init(b, MinOp<pair<signed, signed> >());        block_type.assign(n_block, 0);        for (signed i = 0; i < n_block; i++) {            signed cur = 0;            for (signed j = 0; j < block_size - 1; j++) {                signed ind = i * block_size + j;                if (a[ind] < a[ind + 1]) { cur |= 1 << j; }            }            block_type[i] = cur;        }        lookup_table.assign(1 << (block_size - 1), vector<vector<signed> >(block_size, vector<signed>(block_size + 1)));        for (signed i = 0; i < (1 << (block_size - 1)); i++) {            for (signed j = 0; j < block_size; j++) {                signed res = 0;                signed cur = 0;                signed pos = j;                for (signed k = j + 1; k <= block_size; k++) {                    lookup_table[i][j][k] = pos;                    if (i & (1 << (k - 1))) { cur++; } else { cur--; }                    if (res > cur) {                        pos = k;                        res = cur;                    }                }            }        }    }    signed query(signed l, signed r) {        assert(l < r);        signed lb = l / block_size;        signed rb = r / block_size;        if (lb == rb) { return lb * block_size + lookup_table[block_type[lb]][l % block_size][r % block_size]; }        signed pl = lb * block_size + lookup_table[block_type[lb]][l % block_size][block_size];        signed pr = rb * block_size + lookup_table[block_type[rb]][0][r % block_size];        signed pos = pl;        if (r % block_size > 0 && a[pl] > a[pr]) { pos = pr; }        if (lb + 1 == rb) { return pos; }        signed spv = sparse_table.query(lb + 1, rb).second;        if (a[pos] > a[spv]) { return spv; }        return pos;    }};
template<class T=int> class tree : public undigraph<T> {PMORMQ rmq;    int cnt;    vector<signed> id, in;    bool never = true;    bool never_hld = true;    void dfs(int x, int p, int d, int dis = 0) {        id[cnt] = x;        par[x] = p;        rmq_dep.push_back(d);        disv[x] = dis;        in[x] = cnt++;        forg(gi, g[x]) {            if (t == p) { continue; }            dfs(t, x, d + 1, dis + c);            id[cnt] = x;            rmq_dep.push_back(d);            cnt++;        }    }    void precalc() {        never = false;        cnt = 0;        rmq_dep.clear();        disv.assign(n, 0);        in.assign(n, -1);        id.assign(2 * n - 1, -1);        par.assign(n, -1);        dfs(root, -1, 0);        rmq.init(rmq_dep);
#ifdef _DEBUG
        if(n>=100)return;cerr << "---tree---" << endl;        rep(i, n) {            if (!(i == root || sz(g[i]) > 1))continue;            cerr << i << " -> ";            vi ts;            forg(gi, g[i]) { if (t != par[i])ts.push_back(t); }            rep(i, sz(ts) - 1)cerr << ts[i] << ", ";            if(sz(ts))cerr << ts.back() << endl;        }        cerr << endl;
#endif
    }    int pos;    void hld_build() {        never_hld = false;        if (never)precalc();        g.resize(n);        vid.resize(n, -1);        head.resize(n);        heavy.resize(n, -1);        depth.resize(n);        inv.resize(n);        subl.resize(n);        subr.resize(n);        dfs(root, -1);        t = 0;        dfs_hld(root);
#ifdef _DEBUG
    /*@formatter:on*/
        if (n >= 100)return;
        cerr << "---hld_index---" << endl;
        vi inds;
        rep(i, n) if (sz(g[i]))inds.push_back(i);
        rep(i, sz(inds)) {
            str s = tos(bel(inds[i]));
            cerr << std::right << std::setw(sz(s) + (i ? 1 : 0)) << inds[i];
        }
        cerr << endl;
        rep(i, sz(inds)) { cerr << bel(inds[i]) << " "; }
        cerr << endl << endl;
        cerr << "---hld_edge_index---" << endl;
        fora(e, edges) { if (e.f <= e.t) cerr << e.f << "-" << e.t << " " << bel(e) << endl; }
        cerr << endl << endl;
        cerr << "!!edge or not edge carefull!!" << endl;
        cerr << "!!if(f<t) seg.add(bel(f,t),c) carefully(f,t) (t,f)!!" << endl;
#endif
        /*@formatter:off*/
    }    int dfs(int curr, int prev) {        int sub = 1, max_sub = 0;        heavy[curr] = -1;        forg(i, g[curr]) {            int next = t;            if (next != prev) {                depth[next] = depth[curr] + 1;                int sub_next = dfs(next, curr);                sub += sub_next;                if (max_sub < sub_next) max_sub = sub_next, heavy[curr] = next;            }        }        return sub;    }    int t = 0;    void dfs_hld(int v = 0) {        vid[v] = subl[v] = t;        t++;        inv[subl[v]] = v;        if (0 <= heavy[v]) {            head[heavy[v]] = head[v];            dfs_hld(heavy[v]);        }        forg(i, g[v])if (depth[v] < depth[t])                if (t != heavy[v]) {                    head[t] = t;                    dfs_hld(t);                }        subr[v] = t;    }vector<signed> rmq_dep;
public:
    using undigraph<T>::g;    using undigraph<T>::n;    using undigraph<T>::edges;
    vector<int> disv;
    //部分木の  [左端、右端)  index
    //部分木の辺に加算する場合
    //add(subl[i],subr[i],x)
    //add(sub[i],sub[i+1],-x)
    vector<int> vid, head, heavy, par, depth, inv, subl, subr;
    int root;
    tree(int n_, int root = 0) : undigraph<T>(n_), root(root) { n = n_; }
    int lca(int a, int b) {        if (never)precalc();        int x = in[a];        int y = in[b];        if (x > y) { swap(x, y); }        int pos = rmq.query(x, y + 1);        return id[pos];    }
    int dis(int a, int b) {        if (never)precalc();        int x = in[a];        int y = in[b];        if (x > y) { swap(x, y); }        int pos = rmq.query(x, y + 1);        int p = id[pos];        return disv[a] + disv[b] - disv[p] * 2;    }
    /*O(N) hldを使わず木を普通にたどる*/
    void for_each_l(int u, int v, function<void(int)> fnode) {        int r = lca(u, v);        while (u != r) {            fnode(u);            u = par[u];        }        while (v != r) {            fnode(v);            v = par[v];        }        fnode(r);    }
    void for_each_edge_l/*O(N) 頂点に対しての処理順が可換*/(int u, int v, function<void(edge<int> &)> fedge) {        int r = lca(u, v);        auto sub = [&](int u, int r) {            while (u != r) {                forg(gi, g[u]) {                    if (t == par[u]) {                        fedge(g[u][gi]);                        u = par[u];                        break;                    }                }            }        };        sub(u, r);        sub(v, r);    }
    //Fは半開 (u,v)は木の頂点
    //中ではhldの頂点を見るため、seg木のupdateはhldのindexで行なう
    void for_each_/*[l,r)*/(int u, int v, const function<void(int, int)> &f) {        if (never_hld)hld_build();        while (1) {            if (vid[u] > vid[v]) swap(u, v);            int l = max(vid[head[v]], vid[u]);            int r = vid[v] + 1;            f(l, r);            if (head[u] != head[v]) v = par[head[v]]; else break;        }    }
    void for_each_edge/*[l,r) O(log(N)) 辺を頂点として扱っている 上と同じ感じで使える*/(int u, int v, const function<void(int, int)> &f) {        if (never_hld)hld_build();        while (1) {            if (vid[u] > vid[v]) swap(u, v);            if (head[u] != head[v]) {                int l = vid[head[v]];                int r = vid[v] + 1;                f(l, r);                v = par[head[v]];            } else {                if (u != v) {                    int l = vid[u] + 1;                    int r = vid[v] + 1;                    f(l, r);                }                break;            }        }    }
    int bel(int v) {        /*hld内での頂点番号を返す*/        if (never_hld)hld_build();return vid[v];}
    //下の頂点に辺のクエリを持たせている
    int bel(int f, int t) {        /*辺のクエリを扱うときどの頂点に持たせればいいか(vidを返すのでそのままupd出来る)*/        if (never_hld)hld_build();return depth[f] > depth[t] ? vid[f] : vid[t];}
    int bel(edge<T> &e) {        /*辺のクエリを扱うときどの頂点に持たせればいいか(vidを返すのでそのままupd出来る)*/        if (never_hld)hld_build();return depth[e.f] > depth[e.t] ? vid[e.f] : vid[e.t];}
    template<class ... U> int operator()(U ... args) { return bel(args...); }
    //path l -> r += v
    template<class S> void seg_add(S &seg, int lhei, int rhei, int v) {for_each_(lhei, rhei, [&](int l, int r) { seg.add(l, r, v); });}
    template<class S> void seg_update(S &seg, int lhei, int rhei, int v) {for_each_(lhei, rhei, [&](int l, int r) { seg.update(l, r, v); });}
    template<class S> T seg_get(S &seg, int lhei, int rhei) {        T res = seg.e;        for_each_(lhei, rhei, [&](int l, int r) { res = seg.f(res, seg.get(l, r)); });        return res;    }
    template<class S> void seg_add_edge(S &seg, int lhei, int rhei, int v) { for_each_edge(lhei, rhei, [&](int l, int r) { seg.add(l, r, v); }); }
    template<class S> void seg_update_edge(S &seg, int lhei, int rhei, int v) { for_each_edge(lhei, rhei, [&](int l, int r) { seg.update(l, r, v); }); }
    template<class S> T seg_get_edge(S &seg, int lhei, int rhei) {        T res = seg.e;        for_each_edge(lhei, rhei, [&](int l, int r) { res = seg.f(res, seg.get(l, r)); });        return res;    }
    //単体   edgeは上で処理できる
    template<class S> void seg_add(S &seg, int i, int v) { seg.add(bel(i), v); }
    template<class S> void seg_update(S &seg, int i, int v) { seg.update(bel(i), v); }
    template<class S> T seg_get(S &seg, int i) { return seg[i]; }
    template<class S> void seg_del(S &seg, int i) { seg.del(bel(i)); }

    //部分木iに対するクエリ
    template<class S> void seg_add_sub(S &seg, int i, int v) {if (never_hld)hld_build();seg.add(subl[i], subr[i], v);}
    template<class S> void seg_update_sub(S &seg, int i, int v) {if (never_hld)hld_build();seg.update(subl[i], subr[i], v);}
    template<class S> T seg_get_sub(S &seg, int i, int v) {if (never_hld)hld_build();return seg.get(subl[i], subr[i], v);}
    template<class S> void seg_add_sub_edge(S &seg, int i, int v) {if (never_hld)hld_build();/*iの上の辺が数えられてしまうため、i+1から*/seg.add(subl[i] + 1, subr[i], v);}
    template<class S> void seg_update_sub_edge(S &seg, int i, int v) {if (never_hld)hld_build();/*iの上の辺が数えられてしまうため、i+1から*/seg.update(subl[i] + 1, subr[i], v);}
    template<class S> T seg_get_sub_edge(S &seg, int i, int v) {if (never_hld)hld_build();/*iの上の辺が数えられてしまうため、i+1から*/return seg.get(subl[i] + 1, subr[i], v);}
};
template<typename T> struct LowLink { /*@formatter:off*/   undigraph<T> &g;    vector<int> used, ord, low;    vector<int> articulation;    vector<pair<int, int> > bridge;    LowLink(undigraph<T> &g) : g(g) {}    int dfs(int idx, int k, int par) {        used[idx] = true;        ord[idx] = k++;        low[idx] = ord[idx];        bool is_articulation = false;        int cnt = 0;        forg(gi, g[idx]) {            if (!used[t]) {                ++cnt;                k = dfs(t, k, idx);                low[idx] = min(low[idx], low[t]);                is_articulation |= ~par && low[t] >= ord[idx];                if (ord[idx] < low[t]) bridge.emplace_back(minmax(idx, (int) t));            } else if (t != par) { low[idx] = min(low[idx], ord[t]); }        }        is_articulation |= par == -1 && cnt > 1;        if (is_articulation) articulation.push_back(idx);        return k;    }    virtual void build() {        used.assign(g.n, 0);        ord.assign(g.n, 0);        low.assign(g.n, 0);        int k = 0;        for (int i = 0; i < g.n; i++) { if (!used[i]) k = dfs(i, k, -1); }    }};
template<typename T = int> class scc_un: LowLink<T> { /****************************************************************************/using LL = LowLink<T>;using LL::g;    void dfs(int idx, int par, int &k) {        if (~par && this->ord[par] >= this->low[idx]) bel_[idx] = bel_[par]; else bel_[idx] = k++;        forg(gi, g[idx]) { if (bel_[t] == -1) dfs(t, idx, k); }    }    vi bel_;
public :
    int operator[](const int &k) { return bel_[k]; }
    void build(undigraph<> &comp) {        LL::build();        bel_.assign(this->g.size(), -1);        int k = 0;        for (int i = 0; i < bel_.size(); i++) { if (bel_[i] == -1) dfs(i, -1, k); }        comp.resize(k);        for (auto &e : this->bridge) {            int x = bel_[e.first], y = bel_[e.second];            comp.add(x, y);        }        list.resize(sz(bel_));        rep(i, bel_.size()) { list[bel_[i]].push_back(i); }
#ifdef _DEBUG
        cerr << "---scc_un_index---" << endl;        vi inds;        rep(i, g.n) if (sz(g[i]))inds.push_back(i);        rep(i, sz(inds)) {            str s = tos(bel(inds[i]));            cerr << std::right << std::setw(sz(s) + (i ? 1 : 0)) << inds[i];        }        cerr << endl;        rep(i, sz(inds)) { cerr << bel(inds[i]) << " "; }        cerr << endl << endl;
#endif
    }
    scc_un(undigraph<> &g) : LL(g) {}
    scc_un(undigraph<> &g, undigraph<> &comp) : LL(g) { build(comp); }
    //[i]  := iが所属するグループid
    int bel(int i) { return bel_[i]; }
    int operator()(int i) { return bel_[i]; }
    vector<vi> list; //[gi] := giに所属する頂点集合
};
//(len, init_value)
//(vector<> &) 参照を付けないとおかしくなる
//答えが存在しないことがある場合 index付きで、答えが-1になるかで判定したほうがいい
/*@formatter:off*/
template<class M=int> struct segmax {
    int n;
    vector<M> seg;
    M f(M a, M b) { return a > b ? a : b; }
    const M e = MIN(M);
    segmax(int len, M init_value = MIN(M)) : e(init_value) {        n = 1;        int asz = len;        while (n <= asz)n <<= 1;        seg.assign((n << 1) - 1, e);        rep(i, asz) seg[i + n - 1] = e;        rer(i, n - 2)seg[i] = f(seg[(i << 1) | 1], seg[(i << 1) + 2]);        debu();    }
    segmax(vector<M> &dat, M init_value = MIN(M)) : e(init_value) {        n = 1;        int asz = dat.size();        while (n <= asz)n <<= 1;        seg.assign((n << 1) - 1, e);        rep(i, asz) seg[i + n - 1] = dat[i];        rer(i, n - 2)seg[i] = f(seg[(i << 1) | 1], seg[(i << 1) + 2]);        debu();    }
    void update(int k, M v, bool debug = true) {        int tk = k;        seg[k + n - 1] = v;        k += n - 1;        while (k) {            k = (k - 1) >> 1;            seg[k] = f(seg[(k << 1) | 1], seg[(k << 1) + 2]);        }        if (debug)debup(tk, v);    }
    void del(int k) { update(k, e); }
    void add(int k, int v, bool debug = true) { update(k, v + seg[k + n - 1], debug); }    M query_bottomup(int l, int r, bool debug = true) {        int tl = l, tr = r;        M retl = e;        M retr = e;        l += n - 1;        r += n - 1;        while (l < r) {            /*右なら、次に右上へ移動してしまうので*/            if (!(l & 1)) { retl = f(retl, seg[l]); }            /*右なら、左を取らないと機会を逃す*/            if (!(r & 1)) { retr = f(seg[r - 1], retr); }            l >>= 1;            r = (r - 1) >> 1;        }        M ret = f(retl, retr);        if (debug)debge(tl, tr, ret);        return ret;    }    M query(int a, int b, int k, int l, int r) {        if (r <= a || b <= l)return e;        else if (a <= l && r <= b)return seg[k];        else {            M sl = query(a, b, (k << 1) | 1, l, (l + r) >> 1);            M sr = query(a, b, (k << 1) + 2, (l + r) >> 1, r);            return f(sl, sr);        }    }/*    M get(int a = 0, int b = -1) { return query(a, b == -1 ? n : b, 0, 0, n); }*/
    M get(int a, int b, bool debug = true) { return query_bottomup(a, min(b, n), debug); }
    M get(int b = inf, bool debug = true) { return query_bottomup(0, min(b, n), debug); }
    int operator()(int b = inf) { return get(0, b); }
    int operator()(int a, int b) { return get(a, b); }
    M operator[](int k) { return seg[k + n - 1]; }
    //r)でv以上を満たす最右のindexを返す
    int most_r(int r, M v) {        M d = get(0, r, false);        if (get(0, r, false) < v) {            err("most_r : ", r, ")  >= ", v, " :  ", -1);            debu();            return -1;        }        int l = 0;        while (r - l > 1) {            int m = (l + r) >> 1;            if (get(m, r, false) >= v) { l = m; } else { r = m; }        }        err("most_r : ", r, ")  >= ", v, " :  ", l);        debu();        return l;    }
    int most_r(M v) { return most_r(n, v); }
    //[lで v以上を満たす最左のindexを返す
    int most_l(int l, M v) {        if (get(l, n, false) < v) {            err("most_l : ", "[ " , l, ">=", v, "  :  ", -1);            debu();            return -1;        }        int r = n;        while (r - l > 1) {            int m = (l + r) >> 1;            if (get(l, m, false) >= v) { r = m; } else { l = m; }        }        err("most_l : ", "[ " , l, ">=", v, "  :  ", l);        debu();        return l;    }
    int most_l(M v) { return most_l(0, v); }
#ifdef _DEBUG
    void debu() {        rep(i, min(n, 10ll)) {            int v = operator[](i);            if (v == MIN(ll) || v == MAX(ll)) { cerr << "e "; } else { cerr << v << " "; }        }        cerr << "" << endl << endl;    }    void debup(int k, int v) {        cerr << "upd : [" << k << "] = " << v << endl;        debu();    }    void debge(int l, int r, M v) {        cerr << "get : [" << l << ", " << r << ") = " << v << endl;        debu();    }
#else
    void debu(){;}    void debup(int k,int v){;}    void debge(int l,int r,M v){;}/*@formatter:on*/
#endif
};

/*@formatter:on*/

undigraph<> g(k5);//nにresizeしないとバグる
tree<> comp(0);//縮約後のグラフ をもつ

priority_queue<int> nqt[k5];//大きい順
/*@formatter:on*/
map<int, int> post;
signed main() {
    din(N, M, Q);
    g.ing(N, M);
    scc_un<> sc(g, comp);
    segmax<> seg(k5, -1);
    rep(i, Q) {
        din(t);
        if (t == 1) {
            din(u, w);
            int tu = sc(u - 1);
#define u !not_use!
            post[w] = tu;
            nqt[tu].push(w);
            if (nqt[tu].top() == w) {
                comp.seg_update(seg, tu, w);
            }
            deb(w, tu);
        } else {
            dind(s, t);
            s = sc(s);
            t = sc(t);//ci
            int res = comp.seg_get(seg, s, t);
            if (res != -1) {
                int i = post[res];
                nqt[i].pop();
                if(sz(nqt[i])){
                    comp.seg_update(seg,i,nqt[i].top());
                }else{
                    comp.seg_del(seg, i);
                }
            }
            cout << res << endl;
        }
    }

}