//#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 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

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))

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 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) 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) 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) cerr <<  debugName(x)<<" = "<<x << endl
#define _deb2(x, y) cerr <<  debugName(x)<<" = "<<x<<", "<< debugName(y)<<" = "<<y<< endl
#define _deb3(x, y, z) cerr <<  debugName(x)<<" = "<<x  << ", " <<  debugName(y)<<" = "<<y <<", " debugName(z)<<" = "<<z <<endl
#define _deb4(x, y, z, a) cerr <<  debugName(x)<<" = "<<x <<", " <<   debugName(y)<<" = "<<y <<", " <<  debugName(z)<<" = "<<z <<", " <<  debugName(a)<<" = "<<a<<endl
#define _deb5(x, y, z, a, b) cerr <<  debugName(x)<<" = "<<x <<", " <<   debugName(y)<<" = "<<y <<", " <<  debugName(z)<<" = "<<z <<", " <<  debugName(a)<<" = "<<a<<", " <<  debugName(b)<<" = "<<b<<endl


#define _overloadebug(_1, _2, _3, _4, _5, name, ...) name
#define debug(...) _overloadebug(__VA_ARGS__,_deb5,_deb4,_deb3,_deb2,_deb1)(__VA_ARGS__)
#define deb(...) _overloadebug(__VA_ARGS__,_deb5,_deb4,_deb3,_deb2,_deb1)(__VA_ARGS__)
#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; }

vi divisors(int v) {
    vi res;
    double lim = std::sqrt(v);
    for (int i = 1; i <= lim; ++i) {
        if (v % i == 0) {
            res.pb(i);
            if (i != v / i)res.pb(v / i);
        }
    }
    return res;
}

vb isPrime;
vi primes;

void setPrime() {
    int len = 4010101;
    isPrime.resize(4010101);
    fill(isPrime, true);
    isPrime[0] = isPrime[1] = false;
    for (int i = 2; i <= sqrt(len) + 5; ++i) {
        if (!isPrime[i])continue;
        for (int j = 2; i * j < len; ++j) {
            isPrime[i * j] = false;
        }
    }
    rep(i, len)if (isPrime[i])primes.pb(i);
}

vi factorization(int v) {
    int tv = v;
    vi res;
    if (isPrime.size() == 0)setPrime();
    for (auto &&p :primes) {
        if (v % p == 0)res.push_back(p);
        while (v % p == 0) {
            v /= p;
        }
        if (v == 1 || p * p > tv)break;
    }
    if (v > 1)res.pb(v);
    return res;
}
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;
}
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
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, 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;
}
int N, K, H, W;
vi A, B, C;
//@formatter:off
//よく使うクラス、構造体
struct unionfind {
    vector<ll> par;
    vector<ll> siz;
    vector<ll> es;
    ll n, trees;//連結グループの数(親の種類)
    unionfind(ll n) : n(n), trees(n) {        par.resize(n);        siz.resize(n);        es.resize(n);        for (ll i = 0; i < n; i++) {            par[i] = i;            siz[i] = 1;        }    }
    ll root(ll x) { if (par[x] == x) { return x; } else { return par[x] = root(par[x]); }}
    bool unite(ll x, ll y) {
        x = root(x);
        y = root(y);
        es[x]++;
        if (x == y) return false;
        if (siz[x] > siz[y]) swap(x, y);
        trees--;
        par[x] = y;
        siz[y] += siz[x];
        es[y] += es[x];
        return true;
    }
    bool same(ll x, ll y) { return root(x) == root(y); }
    ll size(ll x) { return siz[root(x)]; }
    ll esize(ll x) { return es[root(x)]; }
    vi sizes(){        vi cou(n);        vi ret;        ret.reserve(n);        rep(i, n){            cou[root (i)]++;        }        rep(i, n){            if(cou[i])ret.push_back(cou[i]);        }        return ret;    }
    //つながりを無向グラフと見なし、xが閉路に含まれるか判定
    bool close(ll x) { return esize(x) >= size(x); }
    vec<vi> sets() {        vi ind(n, -1);        ll i = 0;        vvi(res, trees);        rep(j, n) {            ll r = root(j);            if (ind[r] == -1)ind[r] = i++;            res[ind[r]].push_back(j);        }        rep(i, trees) {            ll r = root(res[i][0]);            if (res[i][0] == r)continue;            rep(j, 1, sz(res[i])) {                if (res[i][j] == r) {                    swap(res[i][0], res[i][j]);                    break;                }            }        }        return res;    }
};
/*@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,ty, id; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, id=ve[gi].id, ty = ve[gi].ty, true); ++gi)

//typeが追加される
#define forg2(gi, ve) for (ll gi = 0,forglim = ve.size(), f, t, c,ty; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c,ty=ve[gi].ty, true); ++gi)
#define fort2(gi, ve) for (ll gi = 0, f, t, c,ty; gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c,ty=ve[gi].ty, true); ++gi)if(t!=p)
template<class T> struct edge { int f, t; T c; int id; int ty; edge(int f, int t, T c = 1,  int ty = -1,int id = -1) : f(f), t(t), c(c), id(id), ty(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> usedv;public :    vector<vector<edge<T>>> g;    vector<edge<T>> edges;    int n;    graph(int n) : n(n) { g.resize(n), usedv.resize(n); }    void clear() { g.clear(), edges.clear(); }    void resize(int n) {        this->n = n;        g.resize(n);        usedv.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 ty ,int id) = 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>::usedv;
    int eid = 0;

    digraph(int n) : graph<T>(n) {}
    void add(int f, int t, T c = 1, int ty = -1,int id = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {
            debugline("digraph add");
            deb(f, t, c, ty,id);
            ole();
        }
        if (id == -1)id = eid++;
        g[f].emplace_back(f, t, c, ty,id);
        edges.emplace_back(f, t, c, ty,id);
    }
    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 ingct(int n,int m, int minus = 1) {  this->resize(n);      rep(i, m) {            int f, t, c,ty;            cin >> f >> t >> c>>ty;            f -= minus;            t -= minus;            ty -= minus;            add(f, t,c,ty);        }    }
    void ingtc(int n,int m, int minus = 1) {  this->resize(n);      rep(i, m) {            int f, t, c,ty;            cin >> f >> t >> ty>>c;            f -= minus;            t -= minus;            ty -= minus;            add(f, t,c,ty);        }    }
    bool used(edge<T> &e) { return usedv[e.id]; }
    bool used(int id) { return usedv[id]; }
    void del(edge<T> &e) { usedv[e.id] =  1; }
    void del(int id) { usedv[id] =  1; }
    void set_edges() {        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;    using graph<T>::usedv;
    int eid = 0;
    undigraph(int n) : graph<T>(n) {}
    // f < t
    void add(int f, int t, T c = 1, int ty = -1, int id = -1) {
        if (!(0 <= f && f < n && 0 <= t && t < n)) {
            debugline("undigraph add");
            deb(f, t, c, ty, id);
            ole();
        }
        if (id == -1)id = eid++;
        g[f].emplace_back(f, t, c, ty, id);
        g[t].emplace_back(t, f, c, ty, id);
        edges.emplace_back(f, t, c, ty, id);//
        edges.emplace_back(t, f, c, ty, id);
    }
    void add(edge<T> &e) {        int f = e.f, t = e.t, ty = e.ty;        T c = e.c;        add(f, t, c, ty);    }
    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);        }    }
    /*@formatter:on*/
    void ingc(int n, int m, int minus = 1) {
        rep(i, m) {
            this->resize(n);
            int f, t, c;
            cin >> f >> t >> c;
            f -= minus;
            t -= minus;
            add(f, t, c);
        }
    }
    /*@formatter:off*/
    void ingct(int n,int m, int minus = 1) {     this->resize(n);   rep(i, m) {            int f, t, c, ty;            cin >> f >> t >> c >> ty;            f -= minus;            t -= minus;            ty -= minus;            add(f, t, c, ty);        }    }
    void ingtc(int n,int m, int minus = 1) {      this->resize(n);  rep(i, m) {            int f, t, c, ty;            cin >> f >> t >> ty >> c;            f -= minus;            t -= minus;            ty -= minus;            add(f, t, c, ty);        }    }    bool used(edge<T> &e) { return usedv[e.id]; }
    bool used(int id) { return usedv[id]; }
    void del(edge<T> &e) { usedv[e.id] = 1; }
    void del(int id) { usedv[id] = 1; }
    void set_edges() {        if (sz(edges))return;        rep(i, n)fora(e, g[i])edges.push_back(e);    }
};
template<class T> vector<T> dijkstra(const graph<T> &g, int s, int init_value = -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*/    for (auto &&d :dis) if (d == initValue)d = init_value;    return dis;}
/*@formatter:on*/
template<class T> vector<vector<T>> dijkstra_all(const graph<T> &g, int init_value = -1) {
    vector<vector<T>> dis(g.n);
    rep(i, g.n) { dis[i] = dijkstra(g, i, init_value); }
    return dis;
}
/*@formatter:off*/
//ret vector(dis,count); 最短経路とその通りを数える
template<class T> auto dijkstra_cou(const graph<T> &g, int s, int init_value = -1) {    if (!(0 <= s && s < g.n)) {        debugline("dijkstra");        deb(s, g.n);        ole();    }    T initValue = MAX(T);    vector<T> dis(g.n, initValue);    vi cou(g.n);    cou[0] = 1;    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;                cou[to] = cou[e.f];                q.emplace(dis[to], to);            } else if (dis[to] == c) {                cou[to] += cou[e.f];            }        }    }    /*基本、たどり着かないなら-1*/    for (auto &&d :dis) if (d == initValue)d = init_value;    return vtop(dis, cou);}
//コストを無限に減らせる := -linf
//たどり着けない := linf
template<class T> vector<T> bell(graph<T> &g, int s) {    if (g.n >= 1e4) {        cout << "bell size too big" << endl;        exit(0);    }    vector<T> res(g.n, linf);    res[s] = 0;    vb can(g.n);    /*頂点から行けない頂点を持つ、辺を消しておく */    fix([&](auto ds, int p, int i) -> void {        if (can[i])return;        can[i] = true;        forg(gi, g[i])if (t != p)ds(i, t);    })(-1, 0);    vector<edge<T>> es;    fora(e, g.edges) { if (can[e.f])es += e; }    rep(i, g.n) {        bool upd = false;        fora(e, es) {            if (res[e.f] != linf && res[e.t] > res[e.f] + e.c) {                upd = true;                res[e.t] = res[e.f] + e.c;            }        }        if (!upd)break;    }    rep(i, g.n * 2) {        bool upd = 0;        fora(e, g.edges) {            if (res[e.f] != linf && res[e.t] != -linf && res[e.t] > res[e.f] + e.c) {                upd = 1;                res[e.t] = -linf;            }        }        if (!upd)break;    }    return res;}
//コストを無限に増やせる := linf
//たどり着けない := -linf
template<class T> vector<T> bell_far(graph<T> &g, int s) {    if (g.n >= 1e4) {        cout << "bell_far size too big" << endl;        exit(0);    }    vector<T> res(g.n, linf);    res[s] = 0;    vb can(g.n);    /*頂点から行けない頂点を持つ、辺を消しておく*/    fix([&](auto ds, int p, int i) -> void {        if (can[i])return;        can[i] = true;        forg(gi, g[i])if (t != p)ds(i, t);    })(-1, 0);    vector<edge<T>> es;    fora(e, g.edges) { if (can[e.f])es += e; }    rep(i, g.n) {        bool upd = false;        fora(e, es) {            if (res[e.f] != linf && res[e.t] > res[e.f] - e.c) {/*-c*/                upd = true;                res[e.t] = res[e.f] - e.c;/*-c*/            }        }        if (!upd)break;    }    rep(i, g.n * 2) {        bool upd = 0;        fora(e, g.edges) {            if (res[e.f] != linf && res[e.t] != -linf && res[e.t] > res[e.f] - e.c) {/*-c*/                upd = 1;                res[e.t] = -linf;            }        }        if (!upd)break;    }    rep(i, g.n)res[i] *= -1;    return res;}
template<class T> vector<vector<T>> warshall(const graph<T> &g, int init_value = -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*/    rep(i, n)rep(j, n) if (dis[i][j] == linf)dis[i][j] = init_value;    return dis;}
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;        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;            dep.push_back(d);            cnt++;        }    }    void precalc() {        never = false;        cnt = 0;        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(dep);
#ifdef _DEBUG
        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] << ", ";            cerr << ts.back() << endl;        }        cerr << endl;
#endif
    }    int pos;    void hld_build() {        never_hld = false;        if (never)precalc();        pos = 0;        vid.assign(n, -1);        head.assign(n, 0);        sub.assign(n, 1);        hvy.assign(n, -1);        hdep.assign(n, 0);        inv.assign(n, 0);        type.assign(n, 0);        build();
#ifdef _DEBUG
        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;
#endif
    }    void build(vector<int> rs = vector<int>(1, 0)) {        int c = 0;        for (int r:rs) {            dfs(r);            bfs(r, c++);        }    }    void dfs(int rt) {        stack<P> st;        hdep[rt] = 0;        st.emplace(rt, 0);        while (!st.empty()) {            int v = st.top().first;            int &i = st.top().second;            if (i < (int) g[v].size()) {                int u = g[v][i++].t;                if (u == par[v]) continue;                hdep[u] = hdep[v] + 1;                st.emplace(u, 0);            } else {                st.pop();                int res = 0;                forg(gi, g[v]) {                    int u = t;                    if (u == par[v]) continue;                    sub[v] += sub[u];                    if (res < sub[u]) res = sub[u], hvy[v] = u;                }            }        }    }    void bfs(int r, int c) {        int &k = pos;        queue<int> q({r});        while (!q.empty()) {            int h = q.front();            q.pop();            for (int i = h; i != -1; i = hvy[i]) {                type[i] = c;                vid[i] = k++;                inv[vid[i]] = i;                head[i] = h;                forg(gi, g[i])if (t != par[i] && t != hvy[i]) q.push(t);            }        }    }    vi vid;
public:
    using undigraph<T>::g;    using undigraph<T>::n;    using undigraph<T>::edges;    using undigraph<T>::usedv;
    vector<int> disv;
    vector<signed> dep, par;
    vector<int> head, sub, hvy, inv, type, hdep/*おそらくグループ内のdep*/;/*vid := bel()*/    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_(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];    }
    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 hdep[f] > hdep[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 hdep[e.f] > hdep[e.t] ? vid[e.f] : vid[e.t];    }
    template<class ... U> int operator()(U ... args) { return bel(args...); }
};;
//辺によりメモリを大量消費ためedgesを消している
//頂点10^6でメモリを190MB(制限の8割)使う
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>::usedv;    int H, W;    int eid = 0;    void add(int f, int t, T c = 1, int ty = -1, int id = -1) {        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("grid_k6 add");            deb(f, t, c, ty, id);            ole();        }        g[f].emplace_back(f, t, c, ty, eid++);        g[t].emplace_back(t, f, c, ty, eid++);    }    int getid(int h, int w) {        if (!ins(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(vector<vector<char>> 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])if (e.f < e.t)edges.push_back(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>::usedv;    int H, W;    int eid = 0;    void add(int f, int t, T c = 1, int ty = -1, int id = -1) {        if (!(0 <= f && f < n && 0 <= t && t < n)) {            debugline("digrid_k6 add");            deb(f, t, c, ty, id);            ole();        }        g[f].emplace_back(f, t, c, ty, eid++);    }    int getid(int h, int w) {        if (!ins(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(vector<vector<char>> 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.push_back(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, digraph<T> &g) {    os << endl << g.n << " " << sz(g.edges) << endl;    fore(gi, g.edges) { os << f << " " << t << " " << c << endl; }    return os;}template<class T> ostream &operator<<(ostream &os, undigraph<T> &g) {    os << endl << g.n << " " << sz(g.edges) << endl;    fore(gi, g.edges) { if (f < t)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.push_back(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.push_back(i);        fora(e, g[i])if (--nyu[e.t] == 0)q.push(e.t);    }    return res.size() == n;}
//閉路がある時linfを返す
template<class T>int longest_path(digraph<T>& g){    vi top;    if(!topo(top,g)){        return linf;    }    int n=sz(top);    vi dp(n,0);    for(auto &&i : top){        forg(gi, g[i]){            chma(dp[t],dp[i]+1);        }    }    return max(dp);}
template<class T>vi longest_path_v(digraph<T>& g){    vi top;    if(!topo(top,g)){        return vi();    }    int n=sz(top);    vi dp(n,0);    vi pre(n,-1);    for(auto &&i : top){        forg(gi, g[i]){            if(chma(dp[t],dp[i]+1)){                pre[t]=i;}}}int s =std::max_element(dp.begin(),dp.end())-dp.begin();vi path;while(s!=-1){path.push_back(s);s=pre[s];}std::reverse(path.begin(),path.end());return path;}
//連結グラフが与えられる 閉路があるか
template<class T> bool close(undigraph<T> &g) {    int n = 0;    int e = 0;    rep(i, g.n) {        if (sz(g[i]))n++;        forg(gi, g[i]) { e++; }    }    return (e >> 1) >= n;}
template<class T> bool close(undigraph<T> &g, int v) {    unionfind uf(g.n);    rep(i, g.n) {        forg(gi, g[i]) {            if (f < t)break;            if (f == t && f == v)return true;            if (uf.same(f, v) && uf.same(t, v))return true;            uf.unite(f, t);        }    }    return false;}template<class T> bool close(digraph<T> &g) {    vi res;    return topo(res, g);}
template<class T> vi indegree(graph<T> &g) {    vi ret(g.size());    rep(i, g.size()) { forg(gi, g[i]) { ret[t]++; }}    return ret;}
template<class T> vi outdegree(graph<T> &g) {    vi ret(g.size());    rep(i, g.size()) { ret[i] = g[i].size(); }    return ret;}
template<class T> digraph<T> rev(digraph<T> &g) {    digraph<T> r(g.n);    rep(i, g.n) { forg(gi, g[i]) { r.add(t, f, c); }}    return r;}
//橋を列挙する (取り除くと連結でなくなる辺)
template<class T> vp bridge(graph<T> &G) {    static bool was;    vp brid;    vi articulation;    vi ord(G.n), low(G.n);    vb vis(G.n);    function<void(int, int, int)> dfs = [&](int v, int p, int k) {        vis[v] = true;        ord[v] = k++;        low[v] = ord[v];        bool isArticulation = false;        int ct = 0;        for (int i = 0; i < G[v].size(); i++) {            if (!vis[G[v][i].t]) {                ct++;                dfs(G[v][i].t, v, k);                low[v] = min(low[v], low[G[v][i].t]);                if (~p && ord[v] <= low[G[v][i].t]) isArticulation = true;                if (ord[v] < low[G[v][i].t]) brid.push_back(make_pair(min(v, G[v][i].t), max(v, G[v][i].t)));            } else if (G[v][i].t != p) { low[v] = min(low[v], ord[G[v][i].t]); }        }        if (p == -1 && ct > 1) isArticulation = true;        if (isArticulation) articulation.push_back(v);    };    int k = 0;    rep(i, G.n) { if (!vis[i]) dfs(i, -1, k); }    sort(brid.begin(), brid.end());    return brid;}
//間接点を列挙する (取り除くと連結でなくなる点)
template<class T> vi articulation(undigraph<T> &G) {    static bool was;    vp bridge;    vi arti;    vi ord(G.n), low(G.n);    vb vis(G.n);    function<void(int, int, int)> dfs = [&](int v, int p, int k) {        vis[v] = true;        ord[v] = k++;        low[v] = ord[v];        bool isArticulation = false;        int ct = 0;        for (int i = 0; i < G[v].size(); i++) {            if (!vis[G[v][i].t]) {                ct++;                dfs(G[v][i].t, v, k);                low[v] = min(low[v], low[G[v][i].t]);                if (~p && ord[v] <= low[G[v][i].t]) isArticulation = true;                if (ord[v] < low[G[v][i].t]) bridge.push_back(make_pair(min(v, G[v][i].t), max(v, G[v][i].t)));            } else if (G[v][i].t != p) { low[v] = min(low[v], ord[G[v][i].t]); }        }        if (p == -1 && ct > 1) isArticulation = true;        if (isArticulation) arti.push_back(v);    };    int k = 0;    rep(i, G.n) { if (!vis[i]) dfs(i, -1, k); }    sort(arti.begin(), arti.end());    return arti;}
#define kansetu articulation
P farthest(undigraph<> &E, int cur, int pre, int d, vi &D) {    D[cur] = d;    P r = {d, cur};    forg(gi, E[cur]) if (t != pre) {            P v = farthest(E, t, cur, d + 1, D);            r = max(r, v);        }    return r;}
//dagでなければ-1を返す
int diameter(digraph<> &g) {    vi per;    if (!topo(per, g))return -1;    int n = g.n;    vi dp(n,1);    fora(v, per) {        forg(gi, g[v]) {            chma(dp[t], dp[f] + 1);        }    }    return max(dp);}
vi diameter(undigraph<> &E) { /* diameter,center*/vi D[3];    D[0].resize(E.size());    D[1].resize(E.size());    auto v1 = farthest(E, 0, 0, 0, D[0]);    auto v2 = farthest(E, v1.second, v1.second, 0, D[0]);    farthest(E, v2.second, v2.second, 0, D[1]);    int i;    rep(i, D[0].size()) D[2].push_back(max(D[0][i], D[1][i]));    return D[2];}
//i d
vp diameter_p(undigraph<> &E) { /* diameter,center*/vector<int> D[3];    D[0].resize(E.size());    D[1].resize(E.size());    auto v1 = farthest(E, 0, 0, 0, D[0]);    auto v2 = farthest(E, v1.second, v1.second, 0, D[0]);    farthest(E, v2.second, v2.second, 0, D[1]);    int i;    vp res(E.size());    rep(i, D[0].size()) { if (D[0][i] > D[1][i])res[i] = mp(D[0][i], v1.second); else res[i] = mp(D[1][i], v2.second); }    return res;}

/*閉路が1つしかない場合、その閉路に含まれる頂点を1としたvectorを返す*/;
//template<class T> vi get_close1(digraph<T> &g) {    int n = g.n;    queue<int> q;    vi d = outdegree(g);    vi res(n, 1);    rep(i, n) {        if (d[i] == 0) {            q += i;            res[i] = 0;        }    }    auto rg = rev(g);    while (q.size()) {        auto now = q.front();        q.pop();        forg(gi, rg[now]) {            if (--d[t] == 0) {                q += t;                res[t] = 0;            }        }    }    return res;};
//閉路パスを一つ返す
//vi close_path(digraph<> &g) {    int n = g.n;    vi state(n);    vi path;    rep(i, n) if (!state[i]) {            if (fix([&](auto dfs, int v) -> bool {                if (state[v]) {                    if (state[v] == 1) {                        path.erase(path.begin(), find(path.begin(), path.end(), v));                        return true;                    }                    return false;                }                path.push_back(v);                state[v] = 1;                forg(gi, g[v]) {                    if (dfs(t))return true;                }                state[v] = -1;                path.pop_back();                return false;            })(i)) {                return path;            }        }    return vi();}
vi close_path_min(digraph<> &g) {    int n = g.n;    vvi(dis, n);    rep(i, n)dis[i] = dijkstra(g, i, linf);    int mind = linf;    int f=0, t=0;    rep(i, n) {        rep(j, n) {            if (i == j)continue;            if (chmi(mind, dis[i][j] + dis[j][i])) {                f = i;                t = j;            }        }    }    vi path;    auto add = [&](int f, int t) {        int now = f;        while (now != t) {            rep(i, n) {                if (dis[now][i] == 1 && dis[f][i] + dis[i][t] == dis[f][t]) {                    path.push_back(i);                    now = i;                    break;                }            }        }    };    add(f, t);    add(t, f);    return path;}
template<class T> int krus(undigraph<T> &g) {    int res = 0;    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res += e.c; }}    return res;}
//idは 00 11 22のようにedgesに持たれている
template<class T> vi krus_id(undigraph<T> &g) {    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    vi res;    fora(e, E) { if (uf.unite(e.f, e.t)) { res.push_back(e.id); }}    return res;}
template<class T> vector<edge<T>> krus_ed(undigraph<T> &g) {    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    vector<edge<T>> res;    fora(e, E) { if (uf.unite(e.f, e.t)) { res.push_back(e); }}    return res;}
template<class T> tree<T> krus_tr(undigraph<T> &g) {    tree<T> res(g.n);    unionfind uf(g.n);    if (sz(g.edges) == 0)g.set_edges();    int i = 0;    auto E = g.edges;    sort(E);    fora(e, E) { if (uf.unite(e.f, e.t)) { res.add(e.f, e.t); }}    return res;}
template<class T> vector<vector<edge<T>>> type_list(digraph<T> &g) {    vector<vector<edge<T>>> res;    rep(i, g.n) { forg2(gi, g[i]) { res[ty].push_back(g[i][gi]); }}    return res;}
template<class T> vector<vector<edge<T>>> type_list(undigraph<T> &g, int types = -1) {    int tn = types;    if (types == -1)tn = g.n;    rep(i, g.n) { forg2(gi, g[i]) { chma(tn, ty); }}    vector<vector<edge<T>>> res(tn + 1);    vi was(g.n);    rep(i, g.n) {        forg2(gi, g[i]) {            if (f < t)res[ty].push_back(g[i][gi]);            else if (f == t && !was[f]) {                res[ty].push_back(g[i][gi]);                was[f] = 1;            }        }    }    return res;}
/*頂点数がkの木を一つ返す サイズが0の木が帰ったら終了*/
//短い版
tree<> g(2 * k5);

signed main() {
    cin >> N;
    g.ingc(N,N-1,0);
    din(Q);

    rep(i, Q){
        //距離が浅くなる2点
    int md=inf;
    int dis =0;
    int no = -1;
        vi p;
//        na(p,3);
        p.resize(3);
        rep(nai,3) {
            cin >> p[nai];
        }

        auto chk =[&](int l,int r,int nu){
            l=p[l],r=p[r];
            int lca = g.lca(l,r);
            if(chmi(md,g.dis(0,lca))){
                dis = g.dis(l,r);
                no =nu;
            }
        };
        chk(0,1,2);
        chk(0,2,1);
        chk(1,2,0);
        int add = inf;
        deb(dis,no);
        rep(j,3){
            if(j==no)con;
            int d = g.dis(p[no],g.lca(p[j],p[no]));
            chmi(add,d);
        }
        deb(add);
        cout << dis+add << endl;
    }




    return 0;
}