Problem:
Score : 600 points

Problem Statement

Given are N points (x_i, y_i) in a two-dimensional plane.

Find the minimum radius of a circle such that all the points are inside or on it.

Constraints

2 \leq N \leq 50

0 \leq x_i \leq 1000

0 \leq y_i \leq 1000

The given N points are all different.

The values in input are all integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print the minimum radius of a circle such that all the N points are inside or on it.

Your output will be considered correct if the absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

2
0 0
1 0

Sample Output 1

0.500000000000000000

Both points are contained in the circle centered at (0.5,0) with a radius of 0.5.

Sample Input 2

3
0 0
0 1
1 0

Sample Output 2

0.707106781186497524

Sample Input 3

10
10 9
5 9
2 0
0 0
2 7
3 3
2 5
10 0
3 7
1 9

Sample Output 3

6.726812023536805158

If the absolute or relative error from our answer is at most 10^{-6}, the output will be considered correct.

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#undef _DEBUG
//#pragma GCC optimize("Ofast")
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace std::chrono;
#define int long long //todo 消したら動かない intの代わりにsignedを使う
#define ll long long
auto start_time = system_clock::now();
auto past_time = system_clock::now();

#define debugName(VariableName) # VariableName
//最大引数がN
#define over2(o1, o2, name, ...) name
#define over3(o1, o2, o3, name, ...) name
#define over4(o1, o2, o3, o4, name, ...) name
#define over5(o1, o2, o3, o4, o5, name, ...) name
#define over6(o1, o2, o3, o4, o5, o6, name, ...) name
#define over7(o1, o2, o3, o4, o5, o6, o7, name, ...) name
#define over8(o1, o2, o3, o4, o5, o6, o7, o8, name, ...) name
#define over9(o1, o2, o3, o4, o5, o6, o7, o8, o9, name, ...) name
#define over10(o1, o2, o3, o4, o5, o6, o7, o8, o9, o10, name, ...) name

/*@formatter:off*/
//[-n, n)にアクセスできる
//また、外部関数resizeに渡せる

//sizeは[-n, n)でnを返す
template<class T> class mvec {    vector<T> v;    int n;public:    mvec() : n(0), v(0) {}    mvec(int n) : n(n), v(n * 2) {}    mvec(int n, T val) : n(n), v(n * 2, val) {}    auto &operator[](int i) { return v[i + n]; }    auto size() { return n; }    void resize(int sn) {        assert(n == 0);        n = sn;        v.resize(sn * 2);    }    auto begin() { return v.begin(); }    auto rbegin() { return v.rbegin(); }    auto end() { return v.end(); }    auto rend() { return v.rend(); }};
//[]でboolは参照を返さないため特殊化が必要
template<> struct mvec<bool> {    vector<bool> v;    int n;    mvec() : n(0), v(0) {}    mvec(int n) : n(n), v(n * 2) {}    mvec(int n, bool val) : n(n), v(n * 2, val) {}    auto operator[](int i) { return v[i + n]; }    auto size() { return v.size(); }    void resize(int sn) {        assert(n == 0);        n = sn;        v.resize(sn * 2);    }    auto begin() { return v.begin(); }    auto rbegin() { return v.rbegin(); }    auto end() { return v.end(); }    auto rend() { return v.rend(); }};
template<class T>ostream &operator<<(ostream &os, mvec<T>& a) {    int spa = 3;    for(auto&&v : a){        spa = max(spa,(int)(to_string(v).size())+1);    }    int n = (int)a.size();    os<<endl;    for (int i = -n; i < n; i++){        int need = spa - ((int)to_string(i).size());        if(i==-n){            need -= min(need,spa - ((int)to_string(a[i]).size()));        }        while(need--){            os<<" ";        }        os<<i;    }    os<<endl;    int i=-n;    for(auto&&v : a){        int need = spa - ((int)to_string(v).size());        if(i==-n){            need -= min(need,spa - ((int)to_string(i).size()));        }        while(need--){            os<<" ";        }        os<<v;        i++;    }    return os;}
#define mv mvec
#define MV mvec
using mvi = mvec<ll>;
using mvb = mvec<bool>;
using mvs = mvec<string>;
using mvd = mvec<double>;
using mvc = mvec<char>;

#define mvvt0(t) mvec<mvec<t>>
#define mvvt1(t,a) mvec<mvec<t>>a
#define mvvt2(t,a, b) mvec<mvec<t>>a(b)
#define mvvt3(t,a, b, c) mvec<mvec<t>> a(b,mvec<t>(c))
#define mvvt4(t,a, b, c, d) mvec<mvec<t>> a(b,mvec<t>(c,d))
#define mvvi(...) over4(__VA_ARGS__,mvvt4,mvvt3,mvvt2 ,mvvt1,mvvt0)(ll,__VA_ARGS__)

template<typename T> mvec<T> make_mv(size_t a) { return mvec<T>(a); }
template<typename T, typename... Ts> auto make_mv(size_t a, Ts... ts) {return mvec<decltype(make_mv<T>(ts...))>(a, make_mv<T>(ts...));}
#define mvni(name, ...) auto name = make_mv<ll>(__VA_ARGS__)


#ifdef _DEBUG
//https://marycore.jp/prog/cpp/class-extension-methods/ 違うかも
template<class T, class A = std::allocator<T>> struct debtor : std::vector<T, A> {
    using std::vector<T, A>::vector;
    template<class U> int deb_v(U a, int v) { return v; }
    template<class U> int deb_v(debtor<U> &a, int v = 0) {cerr << a.size() << " ";return deb_v(a.at(0), v + 1);}
    template<class U> void deb_o(U a) { cerr << a << " "; }
    template<class U> void deb_o(debtor<U> &a) {for (int i = 0; i < min((int) a.size(), 15ll); i++) { deb_o(a[i]); }if ((int) a.size() > 15) { cerr << "..."; }cerr << endl;}
    typename std::vector<T>::reference operator[](typename std::vector<T>::size_type n) {        if (n < 0 || n >= (int) this->size()) {            int siz = (int) this->size();            cerr << "vector size = ";            int dim = deb_v((*this));            cerr << endl;            cerr << "out index at  " << n << endl;            cerr << endl;            if (dim <= 2) { deb_o((*this)); }            exit(0);        }return this->at(n);}
};
#define vector debtor
#endif

#ifdef _DEBUG
//区間削除は出来ない
template<class T> struct my_pbds_tree {    set<T> s;    auto begin() { return s.begin(); }    auto end() { return s.end(); }    auto rbegin() { return s.rbegin(); }    auto rend() { return s.rend(); }    auto empty() { return s.empty(); }    auto size() { return s.size(); }    void clear() { s.clear(); }    template<class U> void insert(U v) { s.insert(v); }template<class U> void operator+=(U v) { insert(v); }    template<class F> auto erase(F v) { return s.erase(v); }    template<class U> auto find(U v) { return s.find(v); }    template<class U> auto lower_bound(U v) { return s.lower_bound(v); }    template<class U> auto upper_bound(U v) { return s.upper_bound(v); }    auto find_by_order(ll k) {        auto it = s.begin();        for (ll i = 0; i < k; i++)it++;        return it;    }    auto order_of_key(ll v) {        auto it = s.begin();        ll i=0;        for (;it != s.end() && *it <v ; i++)it++;        return i;    }};
#define pbds(T) my_pbds_tree<T>
//gp_hash_tableでcountを使えないようにするため
template<class T,class U> struct my_unordered_map{    unordered_map<T,U> m;    my_unordered_map(){};    auto begin(){        return m.begin();    }    auto end(){return m.end();}    auto cbegin(){return m.cbegin();}    auto cend(){return m.cend();}    template<class V>auto erase(V v){return m.erase(v);}    void clear(){m.clear();}    /*countは gp_hash_tableに存在しない*/    /*!= m.end()*/    template<class V>auto find(V v){return m.find(v);}    template<class V>auto & operator [](V n) { return m[n] ;}};
#define unordered_map my_unordered_map
#define umapi unordered_map<ll,ll>
#define umapp unordered_map<P,ll>
#define umapu unordered_map<uint64_t,ll>
#define umapip unordered_map<ll,P>

#else
//umapはunorderd_mapになる
//umapiはgp_hash_table

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

#define umapi __gnu_pbds::gp_hash_table<ll,ll,xorshift>
#define umapp __gnu_pbds::gp_hash_table<P,ll,xorshift>
#define umapu __gnu_pbds::gp_hash_table<uint64_t,ll,xorshift>
#define umapip __gnu_pbds::gp_hash_table<ll,P,xorshift>

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

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



//@マクロ省略系 型,構造
#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>;
using mvp = mvec<P>;
using mvt = mvec<T>;
#define MIN(a) numeric_limits<a>::min()
#define MAX(a) numeric_limits<a>::max()

#define fi first
#define se second
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==
#define el else
#define elf else if
#define upd update
#define sstream stringstream

#define maxq 1
#define minq -1

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

//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 V vector
#define vvt0(t) vector<vector<t>>
#define vvt1(t,a) vector<vector<t>>a
#define vvt2(t,a, b) vector<vector<t>>a(b)
#define vvt3(t,a, b, c) vector<vector<t>> a(b,vector<t>(c))
#define vvt4(t,a, b, c, d) vector<vector<t>> a(b,vector<t>(c,d))

#define vvi(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(ll,__VA_ARGS__)
#define vvb(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(bool,__VA_ARGS__)
#define vvs(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(string,__VA_ARGS__)
#define vvd(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(double,__VA_ARGS__)
#define vvc(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(char,__VA_ARGS__)
#define vvp(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(P,__VA_ARGS__)
#define vvt(...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(T,__VA_ARGS__)
#define vv(type,...) over4(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(type,__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 vn(type,name, ...) auto name = make_v<type>(__VA_ARGS__)

#define PQ priority_queue<ll, vector<ll>, greater<ll> >
#define tos to_string
using mapi = map<ll, ll>;
using mapp = map<P, ll>;
using mapd = map<dou, ll>;
using mapc = map<char, ll>;
using maps = map<str, ll>;
using seti = set<ll>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<ll>;
#define uset unordered_set
#define useti unordered_set<ll,xorshift>
#define mset multiset
#define mseti multiset<ll>
#define umap unordered_map
#define mmap multimap
//任意のマクロサポート用 使う度に初期化する
int index_,v1_,v2_,v3_;

template<class T> struct pq {
    priority_queue<T, vector<T>, greater<T> > q;/*小さい順*/    T su = 0;
    void clear() {        q = priority_queue<T, vector<T>, greater<T> >();        su = 0;    }
    void operator+=(T v) {        su += v;        q.push(v);    }
    T sum() { return su; }
    T top() { return q.top(); }
    void pop() {        su -= q.top();        q.pop();    }
    T poll() {        T ret = q.top();        su -= ret;        q.pop();        return ret;    }
    ll size() { return q.size(); }
};
template<class T> struct pqg {
    priority_queue<T> q;/*大きい順*/    T su = 0;
    void clear() {        q = priority_queue<T>();        su = 0;    }
    void operator+=(T v) {        su += v;        q.push(v);    }
    T sum() { return su; }
    T top() { return q.top(); }
    void pop() {        su -= q.top();        q.pop();    }
    T poll() {        T ret = q.top();        su -= ret;        q.pop();        return ret;    }
    ll size() { return q.size(); }
};
#define pqi pq<ll>
#define pqgi pqg<ll>
//マクロ 繰り返し
//↓@オーバーロード隔離

#define rep1(n) for(ll rep1i = 0,rep1lim=n; rep1i < rep1lim ; ++rep1i)
#define rep2(i, n) for(ll i = 0,rep2lim=n; i < rep2lim ; ++i)
#define rep3(i, m, n) for(ll i = m,rep3lim=n; i < rep3lim ; ++i)
#define rep4(i, m, n, ad) for(ll i = m,rep4lim=n; i < rep4lim ; i+= ad)
//逆順 閉区間
#define rer2(i, n) for(ll i = n; i >= 0 ; i--)
#define rer3(i, m, n) for(ll i = m,rer3lim=n; i >= rer3lim ; i--)
#define rer4(i, m, n, dec) for(ll i = m,rer4lim=n; i >= rer4lim ; i-=dec)
//ループを一つにまとめないとフォーマットで汚くなるため
#define nex_ind1(i) i++
#define nex_ind2(i, j, J) i = (j + 1 == J) ? i + 1 : i, j = (j + 1 == J ? 0 : j + 1)
#define nex_ind3(i, j, k, J, K)i = (j + 1 == J && k + 1 == K) ? i + 1 : i, j = (k + 1 == K) ? (j + 1 == J ? 0 : j + 1) : j, k = (k + 1 == K ? 0 : k + 1)
#define nex_ind4(i, j, k, l, J, K, L) i = (j + 1 == J && k + 1 == K && l + 1 == L) ? i + 1 : i, j = (k + 1 == K && l + 1 == L) ? (j + 1 == J ? 0 : j + 1) : j, k = (l + 1 == L ?(k + 1 == K ? 0 : k + 1) : k), l = l + 1 == L ? 0 : l + 1
#define nex_ind5(i, j, k, l, m, J, K, L, M) i = (j + 1 == J && k + 1 == K && l + 1 == L && m + 1 == M) ? i + 1 : i, j = (k + 1 == K && l + 1 == L && m + 1 == M) ? (j + 1 == J ? 0 : j + 1) : j, k = (l + 1 == L && m + 1 == M ?(k + 1 == K ? 0 : k + 1) : k), l =  m + 1 == M ? l+1 == L ? 0 : l+1 : l,  m = m + 1 == M ? 0 : m + 1

#define repss2(i, I) for (int i = 0; i < I; i++)
#define repss4(i, j, I, J) for (int i = (J ? 0 : I), j = 0; i < I; nex_ind2(i, j, J))
#define repss6(i, j, k, I, J, K) for (int i = (J && K ? 0 : I), j = 0, k = 0; i < I; nex_ind3(i, j, k, J, K))
#define repss8(i, j, k, l, I, J, K, L) for (int i = (J && K && L ? 0 : I), j = 0, k = 0, l = 0; i < I; nex_ind4(i, j, k, l, J, K, L))
#define repss10(i, j, k, l, m, I, J, K, L, M)for (int i = (J && K && L && M ? 0 : I), j = 0, k = 0, l = 0, m = 0; i < I; nex_ind5(i, j, k, l, m, J, K, L, M))

//i,j,k...をnまで見る
#define reps2(i, n) repss2(i, n)
#define reps3(i, j, n) repss4(i, j, n, n)
#define reps4(i, j, k, n) repss6(i, j, k, n, n, n)
#define reps5(i, j, k, l, n) repss8(i, j, k, l, n, n, n, n)

template<class T>void nex_repv2(int&i,int&j,int&I, int&J,vector<vector<T>>&s){    while(1){        j++;        if(j >= J){            j=0;            i++;            if(i < I){                J = (int)s[i].size();            }        }        if(i >= I || J) return;    }}
template<class T> void nex_repv3(int &i, int &j, int &k, int &I, int &J, int &K, vector<vector<vector<T>>> &s) {    while (1) {        k++;        if (k >= K) {            k = 0;            j++;            if (j >= J) {                j = 0;                i++;                if (i >= I)return;            }        }        J = (int) s[i].size();        K = (int) s[i][j].size();        if (J && K) return;    }}

#define repv_2(i,a) repss2(i, sz(a))
//正方形である必要はない
//直前を持つのとどっちが早いか
#define repv_3(i, j, a) for (int I = (int)a.size(), J = (int)a[0].size(), i = 0, j = 0; i < I; nex_repv2(i,j,I,J,a))
//箱状になっている事が要求される つまり[i] 次元目の要素数は一定
#define repv_4(i, j, k, a) for (int I = (int)a.size(), J = (int)a[0].size(), K =(int)a[0][0].size(), i = 0, j = 0, k=0; i < I; nex_repv3(i,j,k,I,J,K,a))
#define repv_5(i, j, k, l, a) repss8(i, j, k, l, sz(a), sz(a[0]), sz(a[0][0]), sz(a[0][0][0]))
#define repv_6(i, j, k, l, m, a) repss10(i, j, k, l, m, sz(a), sz(a[0]), sz(a[0][0]), sz(a[0][0][0]), sz(a[0][0][0][0]))

template<typename T>struct has_rbegin_rend {private:template<typename U>static auto check(U &&obj) -> decltype(std::rbegin(obj), std::rend(obj), std::true_type{});static std::false_type check(...);public:static constexpr bool value = decltype(check(std::declval<T>()))::value;};template<typename T>constexpr bool has_rbegin_rend_v = has_rbegin_rend<T>::value;template<typename Iterator>class Range {public:Range(Iterator &&begin, Iterator &&end) noexcept: m_begin(std::forward<Iterator>(begin)), m_end(std::forward<Iterator>(end)) {}Iterator begin() const noexcept {return m_begin;}Iterator end() const noexcept {return m_end;}private:const Iterator m_begin;const Iterator m_end;};template<typename Iterator>static inline Range<Iterator>makeRange(Iterator &&begin, Iterator &&end) noexcept {return Range<Iterator>{std::forward<Iterator>(begin), std::forward<Iterator>(end)};}template<typename T>static inline decltype(auto) makeReversedRange(const std::initializer_list<T> &iniList) noexcept {return makeRange(std::rbegin(iniList), std::rend(iniList));}template<typename T,typename std::enable_if_t<has_rbegin_rend_v<T>, std::nullptr_t> = nullptr>static inline decltype(auto)makeReversedRange(T &&c) noexcept {return makeRange(std::rbegin(c), std::rend(c));}/* rbegin(), rend()を持たないものはこっちに分岐させて，エラーメッセージを少なくする*/template<typename T,typename std::enable_if<!has_rbegin_rend<T>::value, std::nullptr_t>::type = nullptr>static inline void makeReversedRange(T &&) noexcept {static_assert(has_rbegin_rend<T>::value, "Specified argument doesn't have reverse iterator.");}

#define form1(st) for (auto &&form_it = st.begin(); form_it != st.end(); ++form_it)
#define form3(k, v, st) for (auto &&form_it = st.begin(); form_it != st.end(); ++form_it)
#define form4(k, v, st, r) for (auto &&form_it = st.begin(); form_it != st.end() && (*form_it).fi < r; ++form_it)
#define form5(k, v, st, l, r) for (auto &&form_it = st.lower_bound(l); form_it != st.end() && (*form_it).fi < r; ++form_it)

#define fors1(st) for (auto &&fors_it = st.begin(); fors_it != st.end(); ++fors_it)
#define fors2(v, st) for (auto &&fors_it = st.begin(); fors_it != st.end(); ++fors_it)
#define fors3(v, st, r) for (auto &&fors_it = st.begin(); fors_it != st.end() && (*fors_it) < r; ++fors_it)
#define fors4(v, st, l, r) for (auto &&fors_it = st.lower_bound(l); fors_it != st.end() && (*fors_it) < r; ++fors_it)

#define forslr3(st, a, b) for (auto &&forslr_it = st.begin(); forslr_it != st.end(); ++forslr_it)
#define forslr4(v, st, a, b) for (auto &&forslr_it = st.begin(); forslr_it != st.end(); ++forslr_it)
#define forslr5(v, st, r, a, b) for (auto &&forslr_it = st.begin(); forslr_it != st.end() && (*forslr_it) < r; ++forslr_it)
#define forslr6(v, st, l, r, a, b) for (auto &&forslr_it = st.lower_bound(l); forslr_it != st.end() && (*forslr_it) < r; ++forslr_it)

template<class U> vector<U>to1d(vector<U>& a){return a;}
template<class U> vector<U>to1d(V<V<U>>&a){vector<U> res;for(auto&& a1 : a)for(auto&& a2 : a1)res.push_back(a2);return res;}
template<class U> vector<U>to1d(V<V<V<U>>>&a){vector<U> res;for(auto&& a1 : a)for(auto&& a2 : a1)for(auto&& a3 : a2) res.push_back(a3);return res;}
template<class U> vector<U>to1d(V<V<V<V<U>>>>&a){vector<U> res;for(auto&& a1 : a)for(auto&& a2 : a1)for(auto&& a3 : a2) for(auto&& a4 : a3)res.push_back(a4);return res;}
template<class U> vector<U>to1d(V<V<V<V<V<U>>>>>&a){vector<U> res;for(auto&& a1 : a)for(auto&& a2 : a1)for(auto&& a3 : a2) for(auto&& a4 : a3)for(auto&& a5 : a4)res.push_back(a5);return res;}
template<class U> vector<U>to1d(V<V<V<V<V<V<U>>>>>>&a){vector<U> res;for(auto&& a1 : a)for(auto&& a2 : a1)for(auto&& a3 : a2) for(auto&& a4 : a3)for(auto&& a5 : a4)for(auto&& a6 : a5)res.push_back(a6);return res;}

#define fora_init_2(a, A) ;
#define fora_init_4(a, b, A, B) auto &&a = A[fora_i]; auto &&b = B[fora_i];
#define fora_init_6(a, b, c, A, B, C) auto &&a = A[fora_i]; auto &&b = B[fora_i]; auto &&c = C[fora_i];
#define fora_init_8(a, b, c, d, A, B, C, D) auto &&a = A[fora_i]; auto &&b = B[fora_i]; auto &&c = C[fora_i]; auto && d = D[fora_i];
#define fora_init(...) over8(__VA_ARGS__, fora_init_8, a, fora_init_6, a, fora_init_4, a, fora_init_2)(__VA_ARGS__)

#define forr_init_2(a, A) auto &&a = A[forr_i];
#define forr_init_4(a, b, A, B) auto &&a = A[forr_i]; auto &&b = B[forr_i];
#define forr_init_6(a, b, c, A, B, C) auto &&a = A[forr_i]; auto &&b = B[forr_i]; auto &&c = C[forr_i];
#define forr_init_8(a, b, c, d, A, B, C, D) auto &&a = A[forr_i]; auto &&b = B[forr_i]; auto &&c = C[forr_i]; auto && d = D[forr_i];
#define forr_init(...) over8(__VA_ARGS__, forr_init_8, a, forr_init_6, a, forr_init_4, a, forr_init_2)(__VA_ARGS__)


#define forp_init(k, v, ...) auto &&k = (*forp_it).fi;auto &&v = (*forp_it).se;
#define form_init(k, v, ...) auto &&k = (*form_it).fi;auto &&v = (*form_it).se;
#define fors_init(v, ...) auto &&v = (*fors_it);

#define forlr_init(a,A,ngl,ngr) auto a = A[forlr_i]; auto prev = forlr_i ? A[forlr_i-1] : ngl;auto next = forlr_i+1< rep2lim? A[forlr_i+1] : ngr;
#define forslr_init4(a, A, ngl, ngr) auto a = (*forslr_it); auto prev = (forslr_it!=A.begin())? (*std::prev(forslr_it)) : ngl;auto next = (forslr_it!=std::prev(A.end()))? (*std::next(forslr_it)) : ngr;
#define forslr_init5(a, A, r, ngl, ngr) auto a = (*forslr_it); auto prev = (forslr_it!=A.begin())? (*std::prev(forslr_it)) : ngl;auto next = (forslr_it!=std::prev(A.end()))? (*std::next(forslr_it)) : ngr;
#define forslr_init6(a, A, l, r, ngl, ngr) auto a = (*forslr_it); auto prev = (forslr_it!=A.begin())? (*std::prev(forslr_it)) : ngl;auto next = (forslr_it!=std::prev(A.end()))? (*std::next(forslr_it)) : ngr;
#define forslr_init(...)  over6(__VA_ARGS__,forslr_init6,forslr_init5,forslr_init4)(__VA_ARGS__);

#define fora_2(a, A) for(auto&& a : A)
#define fora_4(a, b, A, B) rep(fora_i, sz(A))
#define fora_6(a, b, c, A, B, C) rep(fora_i, sz(A))
#define fora_8(a, b, c, d, A, B, C, D) rep(fora_i, sz(A))
#define forr_2(a, A) rer(forr_i, sz(A)-1)
#define forr_4(a, b, A, B) rer(forr_i, sz(A)-1)
#define forr_6(a, b, c, A, B, C) rer(forr_i, sz(A)-1)
#define forr_8(a, b, c, d, A, B, C, D) rer(forr_i, sz(A)-1)

//↑@オーバーロード隔離

//rep系はインデックス、for系は中身
#define rep(...) over4(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)
#define rer(...) over4(__VA_ARGS__,rer4,rer3,rer2,)(__VA_ARGS__)
//char用のrep
#define repc(i, m, n) for(char i = m,repc3lim=n; i < repc3lim ; ++i)
//i,j,k...をnまで見る
#define reps(...) over5(__VA_ARGS__,reps5,reps4,reps3,reps2,)(__VA_ARGS__)
#define repss(...) over10(__VA_ARGS__, repss10, a, repss8, a, repss6, a, repss4, a, repss2) (__VA_ARGS__)
//vectorのindexを走査する
//repv(i,j,vvi)
#define repv(...) over6(__VA_ARGS__,repv_6,repv_5,repv_4,repv_3,repv_2,)(__VA_ARGS__)
#define rerv(i, A) for (int i = sz(A)-1; i >= 0 ; i--)
//repvn(dp) nは次元
#define repv1(a) repv(i, a)
#define repv2(a) repv(i, j, a)
#define repv3(a) repv(i, j, k, a)
#define repv4(a) repv(i, j, k, l, a)

#define fora(...) over8(__VA_ARGS__, fora_8, a, fora_6, a, fora_4, a, fora_2)(__VA_ARGS__)
#define forr(...) over8(__VA_ARGS__, forr_8, a, forr_6, a, forr_4, a, forr_2)(__VA_ARGS__)

//#define forr(v, a) for(auto&& v : makeReversedRange(a))
//参照を取らない
#define forv(a, b) for(auto a : to1d(b))

//インデックスを前後含めて走査
#define ring(i, s, len) for (int i = s, prev = (s == 0) ? len - 1 : s - 1, next = (s == len - 1) ? 0 : s + 1, cou = 0; cou < len; cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
//値と前後を見る
#define ringv(v, d) index_=0;for (auto prev = d[sz(d)-1],next= (int)d.size()>1?d[1]:d[0],v = d[0]; index_ < sz(d); index_++, prev = v, v = next, next = (index_>=sz(d)-1?d[0]:d[index_+1]))
// 左右をnext prevで見る 0の左と nの右
#define forlr(v, d, banpei_l, banpei_r) rep(forlr_i,sz(d))

#define form(...) over5(__VA_ARGS__,form5,form4,form3,form2,form1)(__VA_ARGS__)
#define fors(...) over4(__VA_ARGS__,fors4,fors3,fors2,fors1)(__VA_ARGS__)
#define forslr(...) over6(__VA_ARGS__,forslr6,forslr5,forslr4,forslr3)(__VA_ARGS__)
#define forp(k, v, st) for (auto &&forp_it = st.begin(); forp_it != st.end(); ++forp_it)


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

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

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

//埋め込み 存在を意識せずに機能を増やされているもの
namespace std {
    template<> class hash<std::pair<signed, signed>> { public:size_t operator()(const std::pair<signed, signed> &x) const { return hash<ll>()(((ll) x.first << 32) | x.second); }};
    template<> class hash<std::pair<ll, ll>> { public:/*大きいllが渡されると、<<32でオーバーフローするがとりあえず問題ないと判断*/size_t operator()(const std::pair<ll, ll> &x) const { return hash<ll>()(((ll) x.first << 32) | x.second); }};
}
//stream まとめ
/*@formatter:on*/
istream &operator>>(istream &iss, P &a) {
    iss >> a.first >> a.second;
    return iss;
}
template<typename T> istream &operator>>(istream &iss, vector<T> &vec) {
    for (T &x: vec) iss >> x;
    return iss;
}
template<class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {
    os << p.fi << " " << p.se;
    return os;
}
ostream &operator<<(ostream &os, T p) {
    os << p.f << " " << p.s << " " << p.t;
    return os;
}
ostream &operator<<(ostream &os, F p) {
    os << p.a << " " << p.b << " " << p.c << " " << p.d;
    return os;
}
template<typename T> ostream &operator<<(ostream &os, vector<T> &vec) {
    for (ll i = 0; i < vec.size(); ++i)os << vec[i] << (i + 1 == vec.size() ? "" : " ");
    return os;
}

template<typename T> ostream &operator<<(ostream &os, vector<vector<T>> &vec) {
    for (ll i = 0; i < vec.size(); ++i) {
        for (ll j = 0; j < vec[i].size(); ++j) { os << vec[i][j] << " "; }
        os << endl;
    }
    return os;
}
template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {
    for (auto &&v:m) os << v;
    return os;
}
template<class T> ostream &operator<<(ostream &os, set<T> s) {
    fora(v, s) { os << v << " "; }
    return os;
}
template<class T> ostream &operator<<(ostream &os, deque<T> a) {
    fora(v, a)os << v << " ";
    return os;
}
ostream &operator<<(ostream &os, vector<vector<char>> &vec) {
    rep(h, sz(vec)) {
        rep(w, sz(vec[0])) { os << vec[h][w]; }
        os << endl;
    }
    return os;
}
//template<class T,class U>ostream &operator<<(ostream &os, vector<pair<T,U>>& a) {fora(v,a)os<<v<<endl;return os;}
/*@formatter:off*/
template<typename W, typename H> void resize(W &vec, const H head) { vec.resize(head); }
template<typename W, typename H, typename ... T> void resize(W &vec, const H &head, const T ... tail) {vec.resize(head);for (auto &v: vec)resize(v, tail...);}
//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;}
//all_of(A, %2);
//all_of(A, a, a%2);
#define all_of__2(a,right) all_of2(a,lam(right))
#define all_of__3(a,v,siki) all_of2(a,[&](auto v){return siki;})
#define all_of(...) over3(__VA_ARGS__,all_of__3,all_of__2)(__VA_ARGS__)
#define all_of_f(a,f) all_of2(a,f)

#define any_of__2(a,right) any_of2(a,lam(right))
#define any_of__3(a,v, siki) any_of2(a,[&](auto v){return siki;})
#define any_of(...) over3(__VA_ARGS__,any_of__3,any_of__2)(__VA_ARGS__)
#define any_of_f(a,f) any_of2(a,f)

#define none_of__2(a,right) none_of2(a,lam(right))
#define none_of__3(a,v, siki) none_of2(a,[&](auto v){return siki;})
#define none_of(...) over3(__VA_ARGS__,none_of__3,none_of__2)(__VA_ARGS__)
#define none_of_f(a,f) none_of2(a,f)

#define find_if__2(a,right) find_if2(a,lam(right))
#define find_if__3(a,v, siki) find_if2(a,[&](auto v){return siki;})
#define find_if(...) over3(__VA_ARGS__,find_if__3,find_if__2)(__VA_ARGS__)
#define find_if_f(a,f) find_if2(a,f)

#define rfind_if__2(a,right) rfind_if2(a,lam(right))
#define rfind_if__3(a,v, siki) rfind_if2(a,[&](auto v){return siki;})
#define rfind_if(...) over3(__VA_ARGS__,rfind_if__3,rfind_if__2)(__VA_ARGS__)
#define rfind_if_f(a,f) rfind_if2(a,f)

#define contains_if__2(a,right) contains_if2(a,lam(right))
#define contains_if__3(a,v, siki) contains_if2(a,[&](auto v){return siki;})
#define contains_if(...) over3(__VA_ARGS__,contains_if__3,contains_if__2)(__VA_ARGS__)
#define contains_if_f(a,f) contains_if2(a,f)

#define count_if__2(a,right) count_if2(a,lam(right))
#define count_if__3(a,v, siki) count_if2(a,[&](auto v){return siki;})
#define count_if(...) over3(__VA_ARGS__,count_if__3,count_if__2)(__VA_ARGS__)
#define count_if_f(a,f) count_if2(a,f)

#define for_each__2(a,right) do{fora(v,a){v right;}}while(0)
#define for_each__3(a,v, siki) do{fora(v,a){siki;}}while(0)
#define for_each(...) over3(__VA_ARGS__,for_each__3,for_each__2)(__VA_ARGS__)
#define for_each_f(a, f) do{fora(v,a){f(v);}}while(0)

#define erase_if__2(a,right) erase_if2(a,lam(right))
#define erase_if__3(a,v, siki) erase_if2(a,[&](auto v){return siki;})
#define erase_if(...) over3(__VA_ARGS__,erase_if__3,erase_if__2)(__VA_ARGS__)
#define erase_if_f(a,f) erase_if2(a,f)

#define entry_if__2(a,right) erase_if2(a,[&](auto v){return !(v right);})
#define entry_if__3(a,v, siki) erase_if2(a,[&](auto v){return !(siki);})
#define entry_if(...) over3(__VA_ARGS__,entry_if__3,entry_if__2)(__VA_ARGS__)
#define entry_if_f(a,f) erase_if2(a,[&](auto v){return !f(v);})


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

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


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

#define din1(a) ll a;cin>>a
#define din2(a, b) ll a,b;cin>>a>> b
#define din3(a, b, c) ll a,b,c;cin>>a>>b>>c
#define din4(a, b, c, d) ll a,b,c,d;cin>>a>>b>>c>>d
#define din5(a, b, c, d, e) ll a,b,c,d,e;cin>>a>>b>>c>>d>>e
#define din6(a, b, c, d, e, f) ll a,b,c,d,e,f;cin>>a>>b>>c>>d>>e>>f
#define din(...) over6(__VA_ARGS__,din6,din5,din4,din3,din2 ,din1)(__VA_ARGS__)

#define dins1(a) str a;cin>>a
#define dins2(a, b) str a,b;cin>>a>> b
#define dins3(a, b, c) str a,b,c;cin>>a>>b>>c
#define dins4(a, b, c, d) str a,b,c,d;cin>>a>>b>>c>>d
#define dins5(a, b, c, d, e) str a,b,c,d,e;cin>>a>>b>>c>>d>>e
#define dins6(a, b, c, d, e, f) str a,b,c,d,e,f;cin>>a>>b>>c>>d>>e>>f
#define dins(...) over6(__VA_ARGS__,dins6,dins5,dins4,dins3,dins2 ,dins1)(__VA_ARGS__)

#define din1d(a) din1(a);a--
#define din2d(a, b) din2(a,b);a--,b--
#define din3d(a, b, c) din3(a,b,c);a--,b--,c--
#define din4d(a, b, c, d) din4(a,b,c,d);a--,b--,c--,d--
#define dind(...) over4(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)

template<class T> void out2(T &&head) { cout << head; }
template<class T, class... U> void out2(T &&head, U &&... tail) {    cout << head << " ";    out2(tail...);}
template<class T, class... U> void out(T &&head, U &&... tail) {    cout << head << " ";    out2(tail...);    cout << "" << endl;}
template<class T> void out(T &&head) {    cout << head  << endl;}
void out() {    cout << ""  << endl;}


#ifdef _DEBUG
template<class T> 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;}
/*@formatter:on*/
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];
        if (typeid(a[i]) == typeid(P)) {
            ss << endl;
        } else {
            ss << " ";
        }
    }
    return ss.str();
}
/*@formatter:off*/
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();}
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 = 12;    H = min(H, sz(a));    rep(i, H) {        ss << endl;        ss << out_m2(a[i], W, U, i);    }       return ss.str();}
template<class T, size_t N> string out_m2(T (&a)[N]) {    vector<T> b;    resize(b, N);    rep(i, N) {        b[i] = a[i];    }    return out_m2(b);}
template<class T, size_t N, size_t M> string out_m2(T (&a)[N][M]) {    vector<vector<T>> b;    resize(b, N, M);    rep(i, N) {        rep(j, M) {            b[i][j] = a[i][j];        }    }    return out_m2(b);}
template<class T, size_t N, size_t M, size_t O> string out_m2(T (&a)[N][M][O]) {    vector<vector<vector<T>>> b;    resize(b, N, M, O);    rep(i, N) {        rep(j, M) {            rep(k, O) {                b[i][j][k] = a[i][j][k];            }        }    }    return out_m2(b);}
string out_m2(int a) {stringstream ss;ss << a;return ss.str();}
/*@formatter:on*/
template<class T> string out_m2(mvec<mvec<T> > &a, ll H = inf, ll W = inf, int key = inf) {
    H = min({H, sz(a), 6ll});
    W = min({W, sz(a[0]), 6ll});
    stringstream ss;
    ss << endl;
//    if (key == inf)ss << " *|"; else ss << " " << key << "|";
    if (key == inf)ss << " *|";
    else {
        ss << std::right << std::setw(2) << key;
        ss << "|";
    }
    rep(w, -W, W)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    rep(w, W * 8 + 3)ss << "_";
    ss << "" << endl;
    rep(h, -H, H) {
        ss << std::right << std::setw(2) << h << "|";
        int NW = min(sz(a[h]), 6ll);
        rep(w, -NW, NW) { if (abs(a[h][w]) == linf) ss << "   e" << ""; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    return ss.str();
}/*@formatter:on*/
template<class T> string out_m2(mvec<mvec<mvec<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {
    stringstream ss;
    if (H == inf)H = 6;
    H = min(H, sz(a));
    rep(i, -H, H) {
        ss << endl;
        ss << out_m2(a[i], W, U, i);
    }
    return ss.str();
}

/*@formatter:off*/
template<class T> string out_m2(T &a) {stringstream ss;ss << a;return ss.str();}
/*@formatter:on*/
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();
}
/*@formatter:off*/
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();}
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
template<class T> void outl(vector<T> &a, int n = inf) { rep(i, min(n, sz(a)))cout << a[i] << endl; }
//テーブルをスペースなしで出力
template<class T> void outt(vector<vector<T>>&a){    rep(i,sz(a)){        rep(j,sz(a[i])){            cout<<a[i][j];        }        cout<<endl;    }}
//int型をbit表記で出力
void outb(int a){cout << bitset<20>(a) << endl;}
template<class T> void na(vector<T> &a, ll n) {    a.resize(n);    rep(i, n)cin >> a[i];}
#define dna(a, n) vi a(n); rep(dnai,n) cin >> a[dnai];
#define dnad(a, n) vi a(n); rep(dnai,n) cin >> a[dnai],a[dnai]--;
template<class T> void nao(vector<T> &a, ll n) {    a.resize(n + 1);    a[0] = 0;    rep(i, n)cin >> a[i + 1];}
template<class T> void naod(vector<T> &a, ll n) {    a.resize(n + 1);    a[0] = 0;    rep(i, n)cin >> a[i + 1], a[i + 1]--;}
template<class T> void nad(vector<T> &a, ll n) {    a.resize(n);    rep(i, n)cin >> a[i], a[i]--;}
template<class T, class U> void na2(vector<T> &a, vector<U> &b, ll n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i];}
#define dna2(a, b, n) vi a(n),b(n);rep(dna2i, n)cin >> a[dna2i] >> b[dna2i];
template<class T, class U> void nao2(vector<T> &a, vector<U> &b, ll n) {    a.resize(n + 1);    b.resize(n + 1);    a[0] = b[0] = 0;    rep(i, n)cin >> a[i + 1] >> b[i + 1];}
#define dna2d(a, b, n) vi a(n),b(n);rep(dna2di, n){cin >> a[dna2di] >> b[dna2di];a[dna2di]--,b[dna2di]--;}
template<class T, class U> void na2d(vector<T> &a, vector<U> &b, ll n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i], a[i]--, b[i]--;}
template<class T, class U, class W> void na3(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i];}
#define dna3(a, b, c, n) vi a(n),b(n),c(n);   rep(dna3i, n)cin >> a[dna3i] >> b[dna3i] >> c[dna3i];
template<class T, class U, class W> void na3d(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;}
#define dna3d(a, b, c, n) vi a(n),b(n),c(n);  rep(dna3di, n){cin >> a[dna3di] >> b[dna3di] >> c[dna3di];a[dna3di]--,b[dna3di]--,c[dna3di]--;}
template<class T, class U, class W, class X> void na4(vector<T> &a, vector<U> &b, vector<W> &c, vector<X> &d, ll n) {a.resize(n);    b.resize(n);    c.resize(n);    d.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i] >> d[i];}
#define dna4(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   rep(dna4i, n)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i];
#define dna4d(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   rep(dna4i, n)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i],--a[dna4i] ,-- b[dna4i],-- c[dna4i],--d[dna4i];
#define nt(a, h, w) resize(a,h,w);rep(nthi,h)rep(ntwi,w) cin >> a[nthi][ntwi];
#define ntd(a, h, w) resize(a,h,w);rep(ntdhi,h)rep(ntdwi,w) cin >> a[ntdhi][ntdwi], a[ntdhi][ntdwi]--;
#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');rep(ntphi,1,h+1)rep(ntpwi,1,w+1) cin >> a[ntphi][ntpwi];
//デバッグ
#define sp << " " <<

#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__)


#ifdef _DEBUG
#define deb(...)  cerr<< over10(__VA_ARGS__,deb10,deb9,deb8,deb7,deb6,deb5,deb4,deb3,deb2,deb1)(__VA_ARGS__) <<endl
#else
#define deb(...) ;
#endif


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


//@formatter:off
//よく使うクラス、構造体
//graphでredefineしているため、書き換えたら向こうも書き換える
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]); }}
    ll operator()(ll x){return root(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); }
    vector<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


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

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


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

//便利関数

//テスト用
#define rand xor128_
unsigned long xor128_(void){static unsigned long x=123456789,y=362436069,z=521288629,w=88675123;unsigned long t;t=(x^(x<<11));x=y;y=z;z=w; return( w=(w^(w>>19))^(t^(t>>8)) );}
char ranc() { return (char) ('a' + rand() % 26); }
ll rand(ll min, ll max) {    assert(min <= max);    if (min >= 0 && max >= 0) { return rand() % (max + 1 - min) + min; } else if (max < 0) { return -rand(-max, -min); } else { if (rand() % 2) { return rand(0, max); } else { return -rand(0, -min); }}}
vi ranv(ll n, ll min, ll max) {    vi v(n);    rep(i, n)v[i] = rand(min, max);    return v;}
str ransu(ll n) {    str s;    rep(i, n)s += (char) rand('A', 'Z');    return s;}
str ransl(ll n) {    str s;    rep(i, n)s += (char) rand('a', 'z');    return s;}
//単調増加
vi ranvinc(ll n, ll min, ll max) {    vi v(n);    bool bad = 1;    while (bad) {        bad = 0;        v.resize(n);        rep(i, n) {            if (i && min > max - v[i - 1]) {                bad = 1;                break;            }            if (i)v[i] = v[i - 1] + rand(min, max - v[i - 1]); else v[i] = rand(min, max);        }    }    return v;}
//便利 汎用

void ranvlr(ll n, ll min, ll max, vi &l, vi &r) {    l.resize(n);    r.resize(n);    rep(i, n) {        l[i] = rand(min, max);        r[i] = l[i] + rand(0, max - l[i]);    }}
template<class T>vector<pair<T,int>> run_length(vector<T> &a) {vector<pair<T,int>> ret;ret.eb(a[0], 1);rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}return ret;}
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<class F> ll mgr(ll ok, ll ng, F f) {
    bool han = true;
    if (ok < ng)
        while (ng - ok > 1) {
            ll mid = (ok + ng) >> 1;
            if (f(mid))ok = mid, han = true; else ng = mid, han = false;
            deb(mid,han);
        }
    else
        while (ok - ng > 1) {
            ll mid = (ok + ng) >> 1;
            if (f(mid))ok = mid, han = true; else ng = mid, han = false;
            deb(mid,han);
        }
    return ok;
}
template<class F> dou mgrd(dou ok, dou ng, F f) {
    bool han = true;
    if (ok < ng)
        rep(i,100) {
            dou mid = (ok + ng) /2;
            if (f(mid))ok = mid, han = true; else ng = mid, han = false;
            if(i<30)deb(mid,han);
        }
    else
        rep(i,100) {
            dou mid = (ok + ng) /2;
            if (f(mid))ok = mid, han = true; else ng = mid, han = false;
            if(i<30)deb(mid,han);
        }
    return ok;
}
//strを整数として比較
string smax(str &a, str b) {    if (sz(a) < sz(b)) { return b; }    else if (sz(a) > sz(b)) { return a; }    else if (a < b)return b;    else return a;}
//strを整数として比較
string smin(str &a, str b) {    if (sz(a) > sz(b)) { return b; }    else if (sz(a) < sz(b)) { return a; }    else if (a > b)return b;    else return a;}
//エラー-1
template<typename W, typename T> ll find(vector<W> &a, int l, const T key) {rep(i, l, sz(a))if (a[i] == key)return i;return -1;}
template<typename W, typename T> ll find(vector<W> &a, const T key) {    rep(i, sz(a))if (a[i] == key)return i;    return -1;}
template<typename W, typename T> P find(vector<vector<W >> &a, const T key) {    rep(i, sz(a))rep(j, sz(a[0]))if (a[i][j] == key)return mp(i, j);    return mp(-1, -1);}
template<typename W, typename U> T find(vector<vector<vector<W >>> &a, const U key) {    rep(i, sz(a))rep(j, sz(a[0]))rep(k, sz(a[0][0]))if (a[i][j][k] == key)return mt(i, j, k);    return mt(-1, -1, -1);}
//stringも書く
int find(string& s, const string key){    int klen = sz(key);    rep(i,sz(s)-klen+1){        if(s[i]!=key[0])continue;        if(s.substr(i,klen) == key){            return i;        }    }    return -1;}
int find(string& s, int l, const string key){    int klen = sz(key);    rep(i, l, sz(s)-klen+1){        if(s[i]!=key[0])continue;        if(s.substr(i,klen) == key){            return i;        }    }    return -1;}
int find(string& s, const char key){    rep(i,sz(s)){        if(s[i]==key)return i;    }    return -1;}
int find(string& s, int l, const char key){    rep(i, l, sz(s)){        if(s[i]==key)return i;    }    return -1;}
template<typename W, typename T> ll count2(W &a, const T k) { return a == k; }
template<typename W, typename T> ll count2(vector<W> &a, const T k) {    ll ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
template<typename W, typename T> ll count(vector<W> &a, const T k) {    ll ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
vi count(vi &a) {    int ma = 0;    fora(v, a) {        if (ma < v)ma = v;    }    vi res(ma + 1);    fora(v, a) {        res[v]++;    }    return res;}
ll count(str &a, str k) {    ll ret = 0, len = k.length();    auto pos = a.find(k);    while (pos != string::npos)pos = a.find(k, pos + len), ++ret;    return ret;}
vi count(str &a) {    vi cou(26);    char c = 'a';    if ('A' <= a[0] && a[0] <= 'Z')c = 'A';    rep(i, sz(a))++cou[a[i] - c];    return cou;}
#define couif count_if
//algorythm

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

ll get(ll a, ll keta) { return (a / (ll) pow(10, keta)) % 10; }
//0は0桁
ll keta(ll v) {    if (v < p10[9]) { if (v < p10[4]) { if (v < p10[2]) { if (v < p10[1]) { if (v < p10[0])return 0; else return 1; } else return 2; } else { if (v < p10[3]) return 3; else return 4; }} else { if (v < p10[7]) { if (v < p10[5]) return 5; else if (v < p10[6])return 6; else return 7; } else { if (v < p10[8])return 8; else return 9; }}}    else {        if (v < p10[13]) { if (v < p10[11]) { if (v < p10[10]) return 10; else return 11; } else { if (v < p10[12]) return 12; else return 13; }}        else {            if (v < p10[15]) {                if (v < p10[14]) return 14;                else return 15;            } else {                if (v < p10[17]) {                    if (v < p10[16]) return 16;                    else return 17;                } else {                    if (v < p10[18])return 18;                    else return 19;                }            }        }    }}
ll dsum(ll v,ll sin=10) {    ll ret = 0;    for (; v; v /= sin)ret += v % sin;    return ret;}

ll mask10(ll v) { return p10[v] - 1; }
//変換系
//[v] := iとなるようなvectorを返す
//存在しない物は-1
template<class T> auto keys(T& a) {    vector<decltype((a.begin())->fi)> res;    for (auto &&k :a)res.push_back(k.fi);    return res;}
template<class T> auto values(T& a) {    vector<decltype((a.begin())->se)> res;    for (auto &&k :a)res.push_back(k.se);    return res;}
template<class T, class U>  bool chma(T &a, const U &b) {    if (a < b) {        a = b;        return true;    }    return false;}
template<class T, class U>  bool chmi(T &a, const U &b) {    if (b < a) {        a = b;        return true;    }    return false;}
template<class T> constexpr T min(T a, signed b) { return a < b ? a : b; }
template<class T> constexpr  T max(T a, signed b) { return a < b ? b : a; }
template<class T> constexpr  T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }
template<class T> constexpr  T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }
template<class T>  T min(vector<T>& a) { return *min_element(all(a)); }
template<class T>  T mini(vector<T>& a) { return min_element(all(a)) - a.begin(); }
template<class T>  T min(vector<T>& a, ll n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T min(vector<T>& a, ll s, ll n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a) { return *max_element(all(a)); }
template<class T,class U>  T max(vector<T>& a,vector<U>& b) { return max(*max_element(all(a)),*max_element(all(b))); }
template<class T>  T maxi(vector<T>& a) { return max_element(all(a)) - a.begin(); }
template<class T>  T max(vector<T>& a, ll n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a, ll s, ll n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<typename A, size_t N> A max(A (&a)[N]) {    A res = a[0];    rep(i, N)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O> A max(A (&a)[N][O]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> A max(A (&a)[N][O][P]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> A max(A (&a)[N][O][P][Q], const T &v) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A max(A (&a)[N][O][P][Q][R]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A max(A (&a)[N][O][P][Q][R][S]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}
template<typename A, size_t N> A min(A (&a)[N]) {    A res = a[0];    rep(i, N)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O> A min(A (&a)[N][O]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> A min(A (&a)[N][O][P]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> A min(A (&a)[N][O][P][Q], const T &v) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A min(A (&a)[N][O][P][Q][R]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A min(A (&a)[N][O][P][Q][R][S]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}
template<class T> T sum(vector<T> &v, ll s, ll t) {    T ret = 0;    rep(i, s, min(sz(v), t))ret += v[i];    return ret;}
template<class T> T sum(vector<T> &v, ll t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<T> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<T> > &v, int t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<vector<T>> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<vector<T>> > &v, int t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<vector<vector<T>>> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<vector<vector<T>>> > &v, int t = inf) { return sum(v, 0, t); }
template<class T> T sum(vector<vector<vector<vector<vector<T>>>> > &v, int s, int t) {T ret = 0;rep(i, s, min(sz(v), t))ret += sum(v[i]);return ret;}
template<class T> T sum(vector<vector<vector<vector<vector<T>>>> > &v, int t = inf) { return sum(v, 0, t); }

template<class T> auto sum(priority_queue<T, vector<T>, greater<T> > &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}
template<class T> auto sum(priority_queue<T> &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}
//template<class T, class U, class... W>  auto sumn(vector<T> &v, U head, W... tail) {    auto ret = sum(v[0], tail...);    rep(i, 1, min(sz(v), head))ret += sum(v[i], tail...);    return ret;}
vi v_i(vi &a) {    int n = max(a) + 1;    vi ret(n, -1);    rep(i, sz(a)) {        ret[a[i]] = i;    }    return ret;}
void clear(PQ &q) { q = PQ(); }
void clear(priority_queue<int>&q) { q = priority_queue<int>();}
template<class T> void clear(queue<T> &q) { while (q.size())q.pop(); }
template<class T> T *negarr(ll size) {    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));    return body + size;}
template<class T> T *negarr2(ll h, ll w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (ll i = 1; i <= 2 * h + 1; ++i) { dummy1[i] = dummy1[i - 1] + 2 * w + 1; }    double **a = dummy1 + h;    return a;}
//imoは0-indexed
//ruiは1-indexed
template<class T> vector<T> imo(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];    return ret;}
//kと同じものの数
template<class T, class U> vi imo(vector<T> &a, U k) {vector<T> ret = a;rep(i, sz(ret))ret[i] = a[i] == k;rep(i, sz(ret) - 1)ret[i + 1] += ret[i];return ret;}
template<class T> vector<T> imox(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}
//漸化的に最小を持つ
template<class T> vector<T> imi(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}
template<class T> vector<T> ima(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chma(ret[i + 1], ret[i]);    return ret;}
//template<class T> vector<T> rimi(vector<T> &v) {    vector<T> ret = v;    rer(i, sz(ret) - 1, 1)chmi(ret[i - 1], ret[i]);    return ret;}
//template<class T> vector<T> rima(vector<T> &v) {    vector<T> ret = v;    rer(i, sz(ret) - 1, 1)chma(ret[i - 1], ret[i]);    return ret;}
template<class T> struct ruiC {
    vector<T> rui;
    ruiC(vector<T> &ru) : rui(ru) {}
    /*先頭0*/
    ruiC() : rui(1, 0) {}
    T operator()(ll l, ll r) {        if (l > r) {            cerr << "ruic ";            deb(l, r);            assert(0);        }        return rui[r] - rui[l];    }
    T operator()(int r) { return operator()(0, r); }
    /*ruiv[]をruic[]に変えた際意味が変わるのがまずいため()と統一*/
    /*単体iを返す 累積でないことに注意(seg木との統一でこうしている)*/
//    T operator[](ll i) { return rui[i + 1] - rui[i]; }
    T operator[](ll i) { return rui[i]; }
    /*0から順に追加される必要がある*/
    void operator+=(T v) { rui.push_back(rui.back() + v); }
    void add(int i, T v) {if (sz(rui) - 1 != i)ole();operator+=(v);}
    T back() { return rui.back(); }
    ll size() { return rui.size(); }
    auto begin(){return rui.begin();}
    auto end(){return rui.end();}
};
template<class T> struct ruimax {template<typename Monoid>struct SegmentTree {        /*pairで処理*/        int sz;        vector<Monoid> seg;        const Monoid M1 = mp(MIN(T), -1);        Monoid f(Monoid a, Monoid b) {            return max(a, b);        }        void build(vector<T> &a) {            int n = sz(a);            sz = 1;            while (sz < n) sz <<= 1;            seg.assign(2 * sz, M1);            rep(i, n) { seg[i + sz] = mp(a[i], i); }            for (int k = sz - 1; k > 0; k--) {                seg[k] = f(seg[k << 1], seg[(k << 1) | 1]);            }        }        Monoid query(int a, int b) {            Monoid L = M1, R = M1;            for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {                if (a & 1) L = f(L, seg[a++]);                if (b & 1) R = f(seg[--b], R);            }            return f(L, R);        }        Monoid operator[](const int &k) const {            return seg[k + sz];        }    };private:vector<T> ve;SegmentTree<pair<T, int>> seg;vector<T> rv;vector<T> ri;bool build = false;
public:
    int n;
    ruimax(vector<T> &a) : ve(a), n(sz(a)) {        int index = -1;        int ma = MIN(T);        rv.resize(n + 1);        ri.resize(n + 1);        rv[0] = -linf;        ri[0] = -1;        rep(i, n) {            if (chma(ma, a[i])) {                index = i;            }            rv[i + 1] = ma;            ri[i + 1] = index;        }    }
    T operator()(int l, int r) {        if (!(l <= r && 0 <= l && r <= n)) {            deb(l, r, n);            assert(0);        }        if (l == 0) {            return rv[r];        } else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).first;        }    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) {        return operator()(0, r);    }
    T getv(int l, int r) {        return operator()(l, r);    }
    T geti(int l, int r) {        assert(l <= r && 0 <= l && r <= n);        if (l == 0) {            return ri[r];        } else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).second;        }    }
    T geti(int r = inf) { return geti(0, min(r,n)); };
    T getv(int r = inf) { return getv(0, min(r,n)); };
    auto begin(){return rv.begin();}
    auto end(){return rv.end();}
};
template<class T> struct ruimin {template<typename Monoid>    struct SegmentTree {  /*pairで処理*/         int sz;        vector<Monoid> seg;        const Monoid M1 = mp(MAX(T), -1);        Monoid f(Monoid a, Monoid b) {            return min(a, b);        }        void build(vector<T> &a) {            int n = sz(a);            sz = 1;            while (sz < n) sz <<= 1;            seg.assign(2 * sz, M1);            rep(i, n) { seg[i + sz] = mp(a[i], i); }            for (int k = sz - 1; k > 0; k--) {                seg[k] = f(seg[k << 1], seg[(k << 1) | 1]);            }        }        Monoid query(int a, int b) {            Monoid L = M1, R = M1;            for (a += sz, b += sz; a < b; a >>= 1, b >>= 1) {                if (a & 1) L = f(L, seg[a++]);                if (b & 1) R = f(seg[--b], R);            }            return f(L, R);        }        Monoid operator[](const int &k) const {            return seg[k + sz];        }    };private:vector<T> ve;SegmentTree<pair<T, int>> seg;vector<T> rv;vector<T> ri;bool build = false;int n;
public:
    ruimin(vector<T> &a) : ve(a), n(sz(a)) {        int index = -1;        int mi = MAX(T);        rv.resize(n + 1);        ri.resize(n + 1);        rv[0] = linf;        ri[0] = -1;        rep(i, n) {            if (chmi(mi, a[i])) {                index = i;            }            rv[i + 1] = mi;            ri[i + 1] = index;        }    }
    T operator()(int l, int r) {        assert(l <= r && 0 <= l && r <= n);        if (l == 0) {            return rv[r];        } else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).first;        }    }
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) {return operator()(0, r);}
    T getv(int l, int r) {        return operator()(l, r);    }
    T geti(int l, int r) {        assert(l <= r && 0 <= l && r <= n);        if (l == 0) {            return ri[r];        } else {            if (!build)seg.build(ve), build = true;            return seg.query(l, r).second;        }    }
    T geti(int r = inf) { return geti(0, min(r,n)); };
    T getv(int r = inf) { return getv(0, min(r,n)); };
    auto begin(){return rv.begin();}
    auto end(){return rv.end();}
};


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() {return ruiC<T>();}
template<class T> ruiC<T> ruic(vector<T> &a) {    vector<T> ret = ruiv(a);    return ruiC<T>(ret);}
vvi() ruib(vi &a) {    vvi(res, 61, sz(a) + 1);    rep(k, 61) {        rep(i, sz(a)) {            res[k][i + 1] = res[k][i] + ((a[i] >> k) & 1);        }    }    return res;}
vector<ruiC<int>> ruibc(vi &a) {    vector<ruiC<int>> ret(61);    vvi(res, 61, sz(a));    rep(k, 61) {        rep(i, sz(a)) {            res[k][i] = (a[i] >> k) & 1;        }        ret[k] = ruic(res[k]);    }    return 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);}

//h query
template<class T> vector<T> imoh(vector<vector<T>> &v, int w) {vector<T> ret(sz(v));rep(h,sz(ret)){ret[h] = v[h][w];}rep(i, sz(ret) - 1){ret[i + 1] += ret[i];}return ret;}
template<class T> vector<T> ruih(vector<vector<T>> &v, int w) {vector<T> ret(sz(v) + 1);rep(h, sz(v)) { ret[h + 1] = v[h][w]; }rep(i, sz(v)) { ret[i + 1] += ret[i]; }return ret;}
template<class T> ruiC<T> ruihc(vector<vector<T>> &a, int w) {vector<T> ret = ruih(a, w);return ruiC<T>(ret);}

//xor
template<class T> struct ruixC {    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> struct rruiC {
    vector<T> rui;
    int n;
    rruiC(vector<T> &a) : n(sz(a)) {rui.resize(n + 1);rer(i, n - 1) {rui[i] = rui[i + 1] + a[i];}}
    /*[r l)*/
    T operator()(int r, int l) {r++;l++;assert(l <= r && l >= 0 && r <= n);return rui[l] - rui[r];}
    T operator()(int l) {return operator()(n - 1, l);}
    T operator[](int i) {return operator()(i);}
};
template<class T> ostream &operator<<(ostream &os, rruiC<T> a) {fora(v, a.rui)os << v << " ";return os;}

#define rrui rruic
template<class T> rruiC<T> rruic(vector<T> &a) { return rruiC<T>(a); }

//掛け算
template<class T> struct ruimulC {
    vector<T> rv;
    int n;
    ruimulC(vector<T> &a) : rv(a), n(sz(a)) {rv.resize(n + 1);rv[0] = 1;rep(i, n) { rv[i + 1] = a[i] * rv[i]; }}
    ruimulC() : n(0) {rv.resize(n + 1);rv[0] = 1;}
    void operator+=(T v) {rv.push_back(rv.back()*v);n++;}
    T operator()(int l, int r) {assert(l <= r && 0 <= l && r <= n);return rv[r] / rv[l];}
    T operator()(int r = inf) { return operator()(0, min(r, n)); }
    T operator[](int r) { return operator()(0, r); }
    auto begin() { return rv.begin(); }
    auto end() { return rv.end(); }
};
template<class T> ruimulC<T> ruimul(vector<T> &a) { return ruimulC<T>(a); }
template<class T> ruimulC<T> ruimul() {vector<T> a; return ruimulC<T>(a); }
/*@formatter:off*/
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> void inc(pair<T,U> &a, U v = 1) { a.first += v,a.second+=v; }
template<class T, class U> void inc(T &a, U v = 1) { a += v; }
template<class T, class U> void inc(vector<T> &a, U v = 1) { for (auto &u:a)inc(u, v); }
template<class T, class U> void dec(T &a, U v = 1) { a -= v; }
template<class T, class U> void dec(vector<T> &a, U v = 1) { for (auto &u :a)dec(u, v); }
template<class U> void dec(string &a, U v = 1) { for (auto &u :a)dec(u, v); }
template<class T,class U, class W> void dec(vector<T> &a,vector<U> &b, W v = 1) { for (auto &u :a)dec(u, v);for (auto &u :b)dec(u, v); }
template<class T,class U,class W> void dec(vector<T> &a,vector<U> &b,vector<W>&c ) { for (auto &u :a)dec(u, 1);for (auto &u :b)dec(u, 1);for (auto &u :c)dec(u, 1); }
bool ins(ll h, ll w, ll H, ll W) { return h >= 0 && w >= 0 && h < H && w < W; }
bool ins(ll l, ll v, ll r) { return l <= v && v < r; }
template<class T> bool ins(vector<T> &a, ll i, ll j = 0) { return ins(0, i, sz(a)) && ins(0, j, sz(a)); }
#define inside ins
ll u(ll a) { return a < 0 ? 0 : a; }
template<class T> vector<T> u(vector<T> &a) {    vector<T> ret = a;    fora(v, ret)v = u(v);    return ret;}

//添え字を返す
template<class F> ll goldd_l(ll left, ll right, F calc) {    double GRATIO = 1.6180339887498948482045868343656;    ll lm = left + (ll) ((right - left) / (GRATIO + 1.0));    ll rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));    ll fl = calc(lm);    ll fr = calc(rm);    while (right - left > 10) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + (ll) ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    ll minScore = MAX(ll);    ll resIndex = left;    for (ll i = left; i < right + 1; ++i) {        ll score = calc(i);        if (minScore > score) {            minScore = score;            resIndex = i;        }    }    return resIndex;}
template<class F> ll goldt_l(ll left, ll right, F calc) {        double GRATIO = 1.6180339887498948482045868343656;        ll lm = left + (ll) ((right - left) / (GRATIO + 1.0));        ll rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));        ll fl = calc(lm);        ll fr = calc(rm);        while (right - left > 10) {            if (fl > fr) {                right = rm;                rm = lm;                fr = fl;                lm = left + (ll) ((right - left) / (GRATIO + 1.0));                fl = calc(lm);            } else {                left = lm;                lm = rm;                fl = fr;                rm = lm + (ll) ((right - lm) / (GRATIO + 1.0));                fr = calc(rm);            }        }    if (left > right) {        ll l = left;        left = right;        right = l;    }    ll maxScore = MIN(ll);    ll resIndex = left;    for (ll i = left; i < right + 1; ++i) {        ll score = calc(i);        if (maxScore < score) {            maxScore = score;            resIndex = i;        }    }    return resIndex;}
/*loopは200にすればおそらく大丈夫 余裕なら300に*/
template<class F> dou goldd_d(dou left, dou right, F calc, ll loop = 200) {    dou GRATIO = 1.6180339887498948482045868343656;    dou lm = left + ((right - left) / (GRATIO + 1.0));    dou rm = lm + ((right - lm) / (GRATIO + 1.0));    dou fl = calc(lm);    dou fr = calc(rm);    /*200にすればおそらく大丈夫*/    /*余裕なら300に*/    ll k = 141;    loop++;    while (--loop) {        if (fl < fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    return left;}
template<class F> dou goldt_d(dou left, dou right, F calc, ll loop = 200) {    double GRATIO = 1.6180339887498948482045868343656;    dou lm = left + ((right - left) / (GRATIO + 1.0));    dou rm = lm + ((right - lm) / (GRATIO + 1.0));    dou fl = calc(lm);    dou fr = calc(rm);    loop++;    while (--loop) {        if (fl > fr) {            right = rm;            rm = lm;            fr = fl;            lm = left + ((right - left) / (GRATIO + 1.0));            fl = calc(lm);        } else {            left = lm;            lm = rm;            fl = fr;            rm = lm + ((right - lm) / (GRATIO + 1.0));            fr = calc(rm);        }    }    return left;}
//l ~ rを複数の区間に分割し、極致を与えるiを返す time-20 msまで探索
template<class F> ll goldd_ls(ll l, ll r, F calc, ll time = 2000) {    auto lim = milliseconds(time - 20);    ll mini = 0, minv = MAX(ll);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        ll haba = (r - l + k) / k;/*((r-l+1) + k-1) /k*/        ll nl = l;        ll nr = l + haba;        rep(i, k) {            ll ni = goldd_l(nl, nr, calc);            if (chmi(minv, calc(ni))) mini = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return mini;}
template<class F> ll goldt_ls(ll l, ll r, F calc, ll time = 2000) {    auto lim = milliseconds(time - 20);    ll maxi = 0, maxv = MIN(ll);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        ll haba = (r - l + k) / k;/*((r-l+1) + k-1) /k*/        ll nl = l;        ll nr = l + haba;        rep(i, k) {            ll ni = goldt_l(nl, nr, calc);            if (chma(maxv, calc(ni))) maxi = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return maxi;}
template<class F> dou goldd_d_s(dou l, dou r, F calc, ll time = 2000) {    /*20ms余裕を持つ*/    auto lim = milliseconds(time - 20);    dou mini = 0, minv = MAX(dou);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        dou haba = (r - l) / k;        dou nl = l;        dou nr = l + haba;        rep(i, k) {            dou ni = goldd_d(nl, nr, calc);            if (chmi(minv, calc(ni))) mini = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return mini;}
template<class F> dou goldt_d_s(dou l, dou r, F calc, ll time = 2000) {    /*20ms余裕を残している*/    auto lim = milliseconds(time - 20);    dou maxi = 0, maxv = MIN(dou);    /*区間をk分割する*/    rep(k, 1, inf) {        auto s = system_clock::now();        dou haba = (r - l) / k;        dou nl = l;        dou nr = l + haba;        rep(i, k) {            dou ni = goldt_d(nl, nr, calc);            if (chma(maxv, calc(ni))) maxi = ni;            nl = nr;            nr = nl + haba;        }        auto end = system_clock::now();        auto part = duration_cast<milliseconds>(end - s);        auto elapsed = duration_cast<milliseconds>(end - start_time);        if (elapsed + part * 2 >= lim) { break; }    }    return maxi;}
template<class T> T min(vector<vector<T >> &a) {    T res = MAX(T);    rep(i, a.size())chmi(res, *min_element(all(a[i])));    return res;}
template<class T> T max(vector<vector<T >> &a) {    T res = MIN(T);    rep(i, a.size())chma(res, *max_element(all(a[i])));    return res;}
constexpr bool bget(ll m, ll keta) {
    #ifdef _DEBUG
        assert(keta<=62);//オーバーフロー 1^62までしか扱えない
    #endif
    return (m >> keta) & 1; }
//bget(n)次元
//                          NならN-1まで
vector<vi> bget2(vi& a,int keta_size){    vvi(res,keta_size,sz(a));    rep(k,keta_size){        rep(i, sz(a)){            res[k][i]=bget(a[i],k);        }    }    return res;}
vi bget1(vi& a,int keta){    vi res(sz(a));    rep(i, sz(a)){        res[i]=bget(a[i],keta);    }    return res;}
ll bget(ll m, ll keta, ll sinsuu) {    m /= (ll) pow(sinsuu, keta);    return m % sinsuu;}
constexpr ll bit(ll n) {
    #ifdef _DEBUG
        assert(n<=62);//オーバーフロー 1^62までしか扱えない
    #endif
    return (1LL << (n)); }
ll bit(ll n, ll sinsuu) { return (ll) pow(sinsuu, n); }
ll mask(ll n) { return (1ll << n) - 1; }
//aをbitに置きなおす
//{0, 2} -> 101
ll biti(vi&a){    int m=0;    for(auto&& v:a) m|=bit(v);    return m;}
//{1, 1, 0} -> 011
ll bitb(vi&a){    int m=0;    rep(i,sz(a)) if(a[i])m|=bit(i);    return m;}
#define bcou __builtin_popcountll
//最下位ビット
ll lbit(ll n) { return n & -n; }
ll lbiti(ll n) { return log2(n & -n); }
//最上位ビット
ll hbit(ll n) {    n |= (n >> 1);    n |= (n >> 2);    n |= (n >> 4);    n |= (n >> 8);    n |= (n >> 16);    n |= (n >> 32);    return n - (n >> 1);}
ll hbiti(ll n){return log2(hbit(n));}
ll hbitk(ll n) {    ll k = 0;    rer(i, 5) {        ll a = k + (1ll << i);        ll b = 1ll << a;        if (b <= n)k += 1ll << i;    }    return k;}
//初期化は0を渡す
ll nextComb(ll &mask, ll n, ll r) {    if (!mask)return mask = (1LL << r) - 1;    ll x = mask & -mask; /*最下位の1*/    ll y = mask + x; /*連続した下の1を繰り上がらせる*/    ll res = ((mask & ~y) / x >> 1) | y;    if (bget(res, n))return mask = 0; else return mask = res;}
//n桁以下でビットがr個立っているもののvectorを返す
vi bitCombList(ll n, ll r) {    vi res;    ll m = 0;    while (nextComb(m, n, r)) { res.push_back(m); }    return res;}
//masの立ってるindexを見る
#define forbit(i, mas) for (int forbitj = lbit(mas), forbitm = mas, i = log2(forbitj); forbitm; forbitm = forbitj ? forbitm ^ forbitj : 0, forbitj = lbit(forbitm), i = log2(forbitj))

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

template<class T, class U, class W> T lowerBound(vector<T> &a, U v, W banpei) {auto it = lower_bound(a.begin(), a.end(), v);if (it == a.end())return banpei; else return *it;}template<class T, class U, class W> T lowerBound(ruiC<T> &a, U v, W banpei) {return lowerBound(a.rui, v, banpei);}
template<class T, class U, class W> T upperBound(vector<T>& a, U v, W banpei){    auto it = upper_bound(a.begin(),a.end(),v);    if(it==a.end())return banpei;    else return *it;}template<class T, class U, class W> T upperBound(ruiC<T> &a, U v, W banpei) {return upperBound(a.rui, v, banpei);}
template<class T, class U, class W> T rlowerBound(vector<T> &a, U v, W banpei) {    auto it = upper_bound(a.begin(), a.end(), v);    if (it == a.begin())return banpei;    else{        return *(--it);    }}template<class T, class U, class W> T rlowerBound(ruiC<T> &a, U v, W banpei) {return rlowerBound(a.rui, v, banpei);}
template<class T, class U, class W> T rupperBound(vector<T> &a, U v, W banpei) {    auto it = lower_bound(a.begin(), a.end(), v);    if (it == a.begin())return banpei;    else{        return *(--it);    }}template<class T, class U, class W> T rupperBound(ruiC<T> &a, U v, W banpei) {return rupperBound(a.rui, v, banpei);}
#define next2(a) next(next(a))
#define prev2(a) prev(prev(a))

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

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

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

//便利 数学 math
//sub ⊂ top
bool subset(int sub, int top) {return (sub & top) == sub;}
//-180 ~ 180 degree
double atand(double h, double w) {return atan2(h, w) / PI * 180;}
//% -mの場合、最小の正の数を返す
ll mod(ll a, ll m) {if (m < 0) m *= -1;return (a % m + m) % m;}
ll pow(ll a) { return a * a; };
ll fact(ll v) { return v <= 1 ? 1 : v * fact(v - 1); }
dou factd(int v){static vd fact(2,1);    if(sz(fact)<=v){        rep(i,sz(fact),v+1){            fact.push_back(fact.back()*i);        }    }    return fact[v];}

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

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

ll sig0(ll t) { return t <= 0 ? 0 : ((1 + t) * t) >> 1; }
//ll sig(ll s, ll t) { return ((s + t) * (t - s + 1)) >> 1; }
ll sig(ll s, ll t) {if (s > t)swap(s, t);return sig0(t - s) + s * (t - s + 1);}

#define tousa_i tosa_i
#define lower_tousa_i lower_tosa_i
#define upper_tousa upper_tosa
#define upper_tousa_i upper_tosa_i
ll tosa_i(ll st, ll ad, ll v) {    assert(((v - st) % ad) == 0);    return (v - st) / ad;}
ll tosa_s(ll st, ll ad, ll len) {    return st * len + sig0(len - 1) * ad;}

// ax + r (x は非負整数) で表せる整数のうち、v 以上となる最小の整数
ll lower_tosa(ll st, ll ad, ll v) {    if (st >= v) return st;    return (v - st + ad - 1) / ad * ad + st;}
//第何項か
ll lower_tosa_i(ll st, ll ad, ll v) {    if (st >= v) return 0;    return (v - st + ad - 1) / ad;}
ll upper_tosa(ll st, ll ad, ll v) { return lower_tosa(st, ad, v + 1); }
ll upper_tosa_i(ll st, ll ad, ll v) { return lower_tosa_i(st, ad, v + 1); }


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


/*@formatter:off*/

//機能拡張
#define dtie(a, b) int a, b; tie(a, b)
template<class T, class U> string to_string(T a, U b) {    string res = "";    res += a;    res += b;    return res;}
template<class T, class U, class V> string to_string(T a, U b, V c) {    string res = "";    res += a;    res += b;    res += c;    return res;}
template<class T, class U, class V, class W> string to_string(T a, U b, V c, W d) {    string res = "";    res += a;    res += b;    res += c;    res += d;    return res;}
template<class T, class U, class V, class W, class X> string to_string(T a, U b, V c, W d, X e) {    string res = "";    res += a;    res += b;    res += c;    res += d;    res += e;    return res;}


constexpr int bsetlen = k5 * 2;
//constexpr int bsetlen = 5050;
#define bset bitset<bsetlen>
bool operator<(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] < b[i])return true;if (a[i] > b[i])return false;}return false;}
bool operator>(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] > b[i])return true;if (a[i] < b[i])return false;}return false;}
bool operator<=(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] < b[i])return true;if (a[i] > b[i])return false;}return true;}
bool operator>=(bitset<bsetlen> &a, bitset<bsetlen> &b) {rer(i, bsetlen - 1) {if (a[i] > b[i])return true;if (a[i] < b[i])return false;}return true;}
string operator~(string &a) {string res = a;for (auto &&c:res) {if (c == '0')c = '1';else if (c == '1')c = '0';else {cerr << "cant ~" << a << "must bit" << endl;exit(0);}}return res;}
ostream &operator<<(ostream &os, bset& a) {    bitset<10> b;    vi list;    rep(i,bsetlen){        if(a[i])list.push_back(i),b[i]=1;    }    os<<b<<", "<<list;    return os;}
int hbiti(bset&a){rer(i,bsetlen){if(a[i])return i;}return -1;}
#define hk(a, b, c) (a <= b && b < c)

//O(N/64)
bset nap(bset &a, int v) {bset r = a | a << v;return r;}
bset nap(bset &a, bset &v) {bset r = a;rep(i, bsetlen) {if (v[i])r |= a << i;}return r;}


template<class T> void seth(vector<vector<T>> &S, int w, vector<T> &v) {assert(sz(S) == sz(v));assert(w < sz(S[0]));rep(h, sz(S)) { S[h][w] = v[h]; }}

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

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


//未分類
//0,2,1 1番目と2番目の次元を入れ替える
template<class T> auto irekae(vector<vector<vector<T> > > &A, int x, int y, int z) {
#define irekae_resize_loop(a, b, c) resize(res,a,b,c);rep(i,a)rep(j,b)rep(k,c)
    vector<vector<vector<T> > > res;    if (x == 0 && y == 1 && z == 2) {        res = A;    } else if (x == 0 && y == 2 && z == 1) {        irekae_resize_loop(sz(A), sz(A[0][0]), sz(A[0])) {                    res[i][j][k] = A[i][k][j];                }    } else if (x == 1 && y == 0 && z == 2) {        irekae_resize_loop(sz(A[0]), sz(A), sz(A[0][0])) {                    res[i][j][k] = A[j][i][k];                }    } else if (x == 1 && y == 2 && z == 0) {        irekae_resize_loop(sz(A[0]), sz(A[0][0]), sz(A)) {                    res[i][j][k] = A[k][i][j];                }    } else if (x == 2 && y == 0 && z == 1) {        irekae_resize_loop(sz(A[0][0]), sz(A), sz(A[0])) {                    res[i][j][k] = A[j][k][i];                }    } else if (x == 2 && y == 1 && z == 0) {        irekae_resize_loop(sz(A[0][0]), sz(A[0]), sz(A)) {                    res[i][j][k] = A[k][j][i];                }    }    return res;
#undef irekae_resize_loop
}
template<class T> auto irekae(vector<vector<T>>&A,int i=1,int j=0){    vvt(res,sz(A[0]),sz(A));    rep(i,sz(A)){        rep(j,sz(A[0])){            res[j][i]=A[i][j];        }    }    return res;}
//tou分割する
template<typename T> vector<vector<T>> cut(vector<T> &a, int tou = 2) {    int N = sz(a);    vector<vector<T>> res(tou);    int hab = N / tou;    vi lens(tou, hab);    rep(i, N % tou) { lens[tou - 1 - i]++; }    int l = 0;    rep(i, tou) {        int len = lens[i];        int r = l + len;        res[i].resize(len);        std::copy(a.begin() + l, a.begin() + r, res[i].begin());        l = r;    }    return res;}
//長さn毎に分割する
template<typename T> vector<vector<T>> cutn(vector<T> &a, int len) {    int N = sz(a);    vector<vector<T>> res(ceil(N, len));    vi lens(N / len, len);    if (N % len)lens.push_back(N % len);    int l = 0;    rep(i, sz(lens)) {        int len = lens[i];        int r = l + len;        res[i].resize(len);        std::copy(a.begin() + l, a.begin() + r, res[i].begin());        l = r;    }    return res;}
vi inds_(vi &a) {    int n = sz(a);    vb was(n);    vi res(n);    rep(i, n) {        assert(!was[a[i]]);        res[a[i]] = i;        was[a[i]] = true;    }    return res;}

//@起動時
struct initon {
    initon() {
        cin.tie(0);
        ios::sync_with_stdio(false);
        cout.setf(ios::fixed);
        cout.precision(16);
        srand((unsigned) clock() + (unsigned) time(NULL));
    };
} initonv;
#define pre prev
#define nex next

//gra mll pr
//上下左右
const string udlr = "udlr";
string UDLR = "UDLR";//x4と連動 UDLR.find('U') := x4[0]
//右、上が正
constexpr ll h4[] = {1, -1, 0, 0};
constexpr ll w4[] = {0, 0, -1, 1};
constexpr ll h8[] = {0, 1, 0, -1, -1, 1, 1, -1};
constexpr ll w8[] = {1, 0, -1, 0, 1, -1, 1, -1};
    int mei_inc(int h, int w, int H, int W, int i) {while (++i < 4) { if (inside(h + h4[i], w + w4[i], H, W))return i; }return i;}
#define mei(nh, nw, h, w) for (int i = mei_inc(h, w, H, W, -1), nh = i<4? h + h4[i] : 0, nw = i<4? w + w4[i] : 0; i < 4; i=mei_inc(h,w,H,W,i), nh = h+h4[i], nw = w+w4[i])

//グラフ内で #undef getid
//#define getidとしているため、ここを書き直したらgraphも書き直す
#define getid_2(h, w) (h * W + w)
#define getid_1(p) (p.first * W + p.second)
#define getid(...) over2(__VA_ARGS__, getid_2, getid_1) (__VA_ARGS__)
#define getp(id) mp(id / W, id % W)
//#define set_shuffle() std::random_device seed_gen;std::mt19937 engine(seed_gen())
//#define shuffle(a) std::shuffle((a).begin(), (a).end(), engine);
//1980 開始からtime ms経っていたらtrue
bool timeup(int time) {
    auto end_time = system_clock::now();
    auto part = duration_cast<milliseconds>(end_time - start_time);
    auto lim = milliseconds(time);
    return part >= lim;
}
void set_time() { past_time = system_clock::now(); }
//MS型(millisecqnds)で返る
//set_timeをしてからの時間
auto calc_time_milli() {    auto now = system_clock::now();    auto part = duration_cast<milliseconds>(now - past_time);    return part;}
auto calc_time_micro() {    auto now = system_clock::now();    auto part = duration_cast<microseconds>(now - past_time);    return part;}
auto calc_time_nano() {    auto now = system_clock::now();    auto part = duration_cast<nanoseconds>(now - past_time);    return part;}
bool calc_time(int zikan) { return calc_time_micro() >= microseconds(zikan); }
using MS=std::chrono::microseconds;
int div(microseconds a, microseconds b) { return a / b; }
int div(nanoseconds a, nanoseconds b) {    if (b < nanoseconds(1)) { return a / nanoseconds(1); }    int v = a / b;    return v;}
//set_time();
//rep(i,lim)shori

//lim*=time_nanbai();
//rep(i,lim)shoriと使う
//全体でmilliかかっていいときにlimを何倍してもう一回できるかを返す
int time_nanbai(int milli) {
    auto dec = duration_cast<nanoseconds>(past_time - start_time);
    auto part = calc_time_nano();
    auto can_time = nanoseconds(milli * 1000 * 1000);
    can_time -= part;
    can_time -= dec;
    return div(can_time, part);
}
/*@formatter:on*/
//vectorで取れる要素数
//bool=> 1e9 * 8.32
//int => 1e8 * 2.6
//ll  => 1e8 * 1.3
//3次元以上取るとメモリがヤバい
//static配列を使う

vvc (ba);
ll N, M, H, W;
vi A, B, C;

//幾何ライブラリ   *@formatter:off*/

vp ort_next_ = {{1, 0}, {0,  1}, {-1, 0}, {0,  -1}, {1,  0}};
vp ort_prev_ = {{1, 0}, {1,  0}, {0, 1}, {-1,  0}, {0,  -1}};

class D {
public:
    double x, y;
    D(double x=0, double y=0): x(x), y(y) {}
    D(P p): x(p.first), y(p.second){}
    D& operator+=(const D& v) { x += v.x; y += v.y; return *this;}
    D operator+(const D& v) const { return D(*this) += v;}
    D& operator-=(const D& v) { x -= v.x; y -= v.y; return *this;}
    D operator-(const D& v) const { return D(*this) -= v;}
    D& operator*=(double s) { x *= s; y *= s; return *this;}
    D operator*(double s) const { return D(*this) *= s;}
    D& operator/=(double s) { x /= s; y /= s; return *this;}
    D operator/(double s) const { return D(*this) /= s;}
    //内積
    double dot(const D& v) const { return x*v.x + y*v.y;}
    //外積
    double cross(const D& v) const { return x*v.y - v.x*y;}
    //距離の2乗
    double norm2() const { return x*x + y*y;}
    double norm() const { return sqrt(norm2());}
    //単位ベクトルを返す
    D unit(){return (*this)/ norm();}
    // 象限
    int ort() const {        if (abs(x) < eps && abs(y) < eps) return 0;        if (y > 0) return x>0 ? 1 : 2;        else return x<0 ? 3 : 4;    }
    //偏角でソート出来る
    bool operator<(const D& v) const {        int o = ort(), vo = v.ort();        if (o != vo) return o < vo;        return cross(v) > 0;    }
    bool operator>(const D &v) const {        int o = ort(), vo = v.ort();        if (o != vo) return o > vo;        return cross(v) < 0;    }
    //反時計回りで隣接する象限との境界にある、長さ1の点を返す
    //(+,+)で呼ぶと(0, 1)が返る
    D ort_bound_next(){        int i=ort();        assert(i);        return D(ort_next_[i]);    }
    //時計回り
    D ort_bound_prev(){        int i=ort();        assert(i);        return D(ort_prev_[i]);    }
};
using dots = vector<D>;


istream& operator>>(istream& ist, D& v) {ist >> v.x >> v.y; return ist;}
ostream& operator<<(ostream& os, const D& v) {os<<"("<<v.x<<","<<v.y<<")"; return os;}

//@基本計算 取得***********************************************

//外積
double cross(const D &p, const D &q) { return p.x * q.y - p.y * q.x; }
//距離
double dis(const D &p, const D &q){return std::sqrt(pow(p.x-q.x,2) +pow(p.y-q.y,2)) ;}
double dis2(const D &p, const D &q){return (pow(p.x-q.x,2) +pow(p.y-q.y,2)) ;}

//@判定*********************************************************
//線分の交差判定 端点を認めない
bool kousa_mid(D fa,D ta,D fb,D tb) {    dou ax = fa.x;    dou ay = fa.y;    dou bx = ta.x;    dou by = ta.y;    dou cx = fb.x;    dou cy = fb.y;    dou dx = tb.x;    dou dy = tb.y;  dou ta1 = (cx - dx) * (ay - cy) + (cy - dy) * (cx - ax);  dou tb1 = (cx - dx) * (by - cy) + (cy - dy) * (cx - bx);  dou tc1 = (ax - bx) * (cy - ay) + (ay - by) * (ax - cx);  dou td1 = (ax - bx) * (dy - ay) + (ay - by) * (ax - dx);  return tc1 * td1 < 0 && ta1 * tb1 < 0;}
//端点を認める
bool kousa_hasi(D fa,D ta,D fb,D tb) {    dou ax = fa.x;    dou ay = fa.y;    dou bx = ta.x;    dou by = ta.y;    dou cx = fb.x;    dou cy = fb.y;    dou dx = tb.x;    dou dy = tb.y;  dou ta1 = (cx - dx) * (ay - cy) + (cy - dy) * (cx - ax);  dou tb1 = (cx - dx) * (by - cy) + (cy - dy) * (cx - bx);  dou tc1 = (ax - bx) * (cy - ay) + (ay - by) * (ax - cx);  dou td1 = (ax - bx) * (dy - ay) + (ay - by) * (ax - dx);   return tc1 * td1 <= 0 && ta1 * tb1 <= 0;}

//@角度系 中心とするものの引数は右に*****************************

//偏角 [-pi,pi] aを中心としたbの偏角
double arg(D b, D a = D(0, 0)) {b -= a;return atan2(b.y, b.x);}

//aを中心にbを何シータ 反時計回りに回転させたらcになるかを返す
//[0, 2pi)
double ang(D b,D c,D a=D(0,0)){    c-=a;    b-=a;    int ob = b.ort();    int oc = c.ort();    if(!ob || !oc){        cerr<<"同じ点は渡せない"<<endl;        re();    }    double ret=0;    if(ob < oc){        ret += (oc-ob-1) * PI/2;        ret += arg(b.ort_bound_next()) - arg(b);        /*prevが-piではなくpiになる       */ if(oc==3){            ret += arg(c)-(-PI);        }else{            ret += arg(c)- arg(c.ort_bound_prev());        }    }else if(ob>oc){        return PI*2 - ang(c,b,D(0,0));    }else{        if(b > c){            return PI*2 - ang(c,b,D(0,0));        }else{            ret = arg(c) - arg(b);        }    }    return ret;}
//角度 aを中心とする最小角度の絶対値

double uang(D b, D c, D a = D(0, 0)) {            dou ret = ang(b, c, a);            return min(ret, 2 * PI - ret);        }
//bを回転 aを中心に反時計回り
D rot(D b,dou theta,D a=D(0,0)){    b-=a;    dou x=b.x,y=b.y;    return D(cos(theta) * x -sin(theta) * y, sin(theta) * x + cos(theta)*y)+a;}

//@変形***********************************************************
D mid(D& a,D& b){return (a+b)/2;}
D change_norm(D b, double want_norm, D a=D(0,0)){b-=a;b/=b.norm();b*= want_norm;return b+a;}

// 一直線上の3点を含めない
// 全ベクトルが同じ方向でどれを足し合わせるかを考えるときに一直線にすると爆発するhttps://atcoder.jp/contests/abc139/tasks/abc139_f F-Engines
//反時計回り
vector<D> convexhull(vector<D> &ps) {int n = (int) ps.size();    if (n <= 2) {        return ps;    }    vector<D> res(2 * n);    std::sort(ps.begin(), ps.end(), [&](D &l, D &r) {        if (l.x != r.x)return l.x < r.x;        else return l.y < r.y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < eps) {                --k;                if (k < 2) break;            }        }        res[k] = ps[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < eps) {                --k;                if (k < t) break;            }        }        res[k] = ps[i];        ++k;    }    res.resize(k - 1);    return res;}
//インデックスを返す
vi convexhull_i(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) {        if (n == 1)return vi{0};else return vi{0, 1};}vi res(2 * n);vi ind = iota(0, n);    std::sort(ind.begin(), ind.end(), [&](int l, int r) {        if (ps[l].x != ps[r].x)return ps[l].x < ps[r].x;        else return ps[l].y < ps[r].y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < eps) {                --k;                if (k < 2) break;            }        }        res[k] = ind[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < eps) {                --k;                if (k < t) break;            }        }        res[k] = ind[i];        ++k;    }    res.resize(k - 1);    return res;}
// 一直線上の3点を含める
vector<D> convexhull_line(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) { return ps; }    vector<D> res(2 * n);    std::sort(ps.begin(), ps.end(), [&](D &l, D &r) { if (l.x != r.x)return l.x < r.x; else return l.y < r.y; });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < -eps) {                --k;                if (k < 2) break;            }        }        res[k] = ps[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < -eps) {                --k;                if (k < t) break;            }        }        res[k] = ps[i];        ++k;    }    res.resize(k - 1);    return res;}
vi convexhull_line_i(vector<D> &ps) {    int n = (int) ps.size();    if (n <= 2) {        if (n == 1)return vi{0};        else return vi{0, 1};    }    vi res(2 * n);    vi ind = iota(0, n);    std::sort(ind.begin(), ind.end(), [&](int l, int r) {        if (ps[l].x != ps[r].x)return ps[l].x < ps[r].x;        else return ps[l].y < ps[r].y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < -eps) {                --k;                if (k < 2) break;            }        }        res[k] = ind[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < -eps) {                --k;                if (k < t) break;            }        }        res[k] = ind[i];        ++k;    }    res.resize(k - 1);    return res;}/*@formatter:on*/


void solve() {
    in(N);
    dots ds;
    na(ds, N);
    
    auto f = [&](dou len) {
        rep(i, N) {
            rep(j, i + 1, N) {
                D a = ds[i];
                D b = ds[j];
                D m = mid(a, b);
                D l = rot(a, PI / 2, m);
                //ookiku
                //何norでmaとの距離がlenになるか
                auto g = [&](dou nor) {
                    D nl = change_norm(l, nor, m);
                    return dis(a, nl) <= len;
                };
                l = change_norm(l, mgrd(0, dinf, g), m);
                auto r = rot(l, -PI, m);
                if (all_of(ds, d, dis(d, l) <= len)) { return true; }
                if (all_of(ds, d, dis(d, r) <= len)) { return true; }
            }
        }
        return false;
    };
    out(mgrd(dinf,0,f));
}

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

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

signed main() {
    solve();

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


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