// Original C++ solution (slower):
//#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;

#define int long long

#define ll long long



//@formatter:off

#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(int k) {        auto it = s.begin();        for (int i = 0; i < k; i++)it++;        return it;    }    auto order_by_key(int v) {        auto it = s.begin();        int i=0;        for (;it != s.end() && *it <v ; i++)it++;        return i;    }};

#define pbds(T) my_pbds_tree<T>

#else

#define unordered_map __gnu_pbds::gp_hash_table

//find_by_order(k) k番目のイテレーター

//order_of_key(k)  k以上が前から何番目か

#define pbds(U) __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>

#endif

struct xorshift {    static uint64_t splitmix64(uint64_t x) {        x += 0x9e3779b97f4a7c15;        x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;        x = (x ^ (x >> 27)) * 0x94d049bb133111eb;        return x ^ (x >> 31);    }    size_t operator()(uint64_t x) const {        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();        return splitmix64(x + FIXED_RANDOM);    }    size_t operator()(std::pair<ll, ll> x) const {        int v=((x.first) << 32) | x.second;        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();        return splitmix64(v + FIXED_RANDOM);    }};

template<class U, class L> void operator+=(__gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update> &s, L v) { s.insert(v); }

//衝突対策

#define ws wszzzz



template<class A, class B, class C>struct T2 {A f;B s;C t;T2() { f = 0, s = 0, t = 0; }T2(A f, B s, C t) : f(f), s(s), t(t) {}bool operator<(const T2 &r) const {        return f != r.f ? f < r.f : s != r.s ? s < r.s : t < r.t;        /*return f != r.f ? f > r.f : s != r.s ?n s > r.s : t > r.t; 大きい順 */   }    bool operator>(const T2 &r) const {        return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t;        /*return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 小さい順 */   }    bool operator==(const T2 &r) const {        return f == r.f && s == r.s && t == r.t;    }    bool operator!=(const T2 &r) const {        return f != r.f || s != r.s || t != r.t;    }};

template<class A, class B, class C, class D> struct F2 {    A a;    B b;    C c;    D d;    F2() { a = 0, b = 0, c = 0, d = 0; }    F2(A a, B b, C c, D d) : a(a), b(b), c(c), d(d) {}    bool operator<(const F2 &r) const {        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c < r.c : d < r.d;    /*    return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c > r.c : d > r.d;*/    }    bool operator>(const F2 &r) const {        return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c > r.c : d > r.d;/*        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c < r.c : d < r.d;*/    }    bool operator==(const F2 &r) const {        return a == r.a && b == r.b && c == r.c && d == r.d;    }    bool operator!=(const F2 &r) const {        return a != r.a || b != r.b || c != r.c || d != r.d;    }    int operator[](int i) {        assert(i < 4);        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;    }};

typedef T2<int, int, int> T;

typedef F2<int, int, int, int> F;

T mt(int a, int b, int c) {return T(a, b, c);}



//@マクロ省略系 型,構造

#define float double

#define flo double

#define double long double

#define ull unsigned long long

using dou = double;

using itn = int;

using str = string;

using bo= bool;

#define au auto

using P = pair<int, int>;

using pd =pair<dou, dou>;

#define fi first

#define se second

#define beg begin

#define rbeg rbegin

#define con continue

#define bre break

#define brk break

#define is ==

#define elf else if

#define wh while



#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) [&](int& p){return p right;}



//マクロ省略系 コンテナ

using vi = vector<int>;

using vb = vector<bool>;

using vs = vector<string>;

using vd = vector<double>;

using vc = vector<char>;

using vp = vector<P>;

using vt = vector<T>;



#define V vector

#define o_vvt(o1, o2, o3, o4, name, ...) name

#define vvt0(t) V<V<t>>

#define vvt1(t,a) V<V<t>>a

#define vvt2(t,a, b) V<V<t>>a(b)

#define vvt3(t,a, b, c) V<V<t>> a(b,V<t>(c))

#define vvt4(t,a, b, c, d) V<V<t>> a(b,V<t>(c,d))



#define vvi(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(int,__VA_ARGS__)

#define vvb(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(bool,__VA_ARGS__)

#define vvs(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(string,__VA_ARGS__)

#define vvd(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(double,__VA_ARGS__)

#define vvc(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(char,__VA_ARGS__)

#define vvp(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(P,__VA_ARGS__)



template<typename T> vector<T> make_v(size_t a) { return vector<T>(a); }

template<typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));}

#define vni(name, ...) auto name = make_v<int>(__VA_ARGS__)

#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)

#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)

#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)

#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)

#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)



#define PQ priority_queue<int, vector<int>, greater<int> >

#define tos to_string

using mapi = map<int, int>;

using mapp = map<P, int>;

using mapd = map<dou, int>;

using mapc = map<char, int>;

using maps = map<str, int>;

using seti = set<int>;

using setd = set<dou>;

using setc = set<char>;

using sets = set<str>;

using qui = queue<int>;

#define bset bitset

#define uset unordered_set

#define useti unordered_set<int,int,xorshift>

#define mset multiset

#define umap unordered_map

#define umapi unordered_map<int,int,xorshift>

#define umapp unordered_map<P,int,xorshift>

#define mmap multimap



template<class T> struct pq {    priority_queue<T, vector<T>, greater<T> > q;/*小さい順*/    T su = 0;    void clear() {q = priority_queue<T, vector<T>, greater<T> >();su = 0;}    void operator+=(T v) {su += v;q.push(v);}    T sum() {return su;}    T top() {return q.top();}    void pop() {su -= q.top();q.pop();}    T poll() {T ret = q.top();su -= ret;q.pop();return ret;}    int size() {return q.size();}};

template<class T> struct pqg {    priority_queue<T> q;/*大きい順*/    T su = 0;    void clear() {q = priority_queue<T>();su = 0;}    void operator+=(T v) {su += v;q.push(v);}    T sum() {return su;}    T top() {return q.top();}    void pop() {su -= q.top();q.pop();}    T poll() {T ret = q.top();su -= ret;q.pop();return ret;}    int size() {return q.size();}};

#define pqi pq<int>

#define pqgi pqg<int>

//マクロ 繰り返し

#define o_rep(o1, o2, o3, o4, name, ...) name

# define rep1(n) for(int rep1i = 0,rep1lim=n; rep1i < rep1lim ; ++rep1i)

# define rep2(i, n) for(int i = 0,rep2lim=n; i < rep2lim ; ++i)

#define rep3(i, m, n) for(int i = m,rep3lim=n; i < rep3lim ; ++i)

#define rep4(i, m, n, ad) for(int i = m,rep4lim=n; i < rep4lim ; i+= ad)

#define rep(...) o_rep(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)



#define rer2(i, n) for(int i = n; i >= 0 ; i--)

#define rer3(i, m, n) for(int i = m,rer3lim=n; i >= rer3lim ; i--)

#define rer4(i, m, n, dec) for(int i = m,rer4lim=n; i >= rer4lim ; i-=dec)

#define rer(...) o_rep(__VA_ARGS__,rer4,rer3,rer2,)(__VA_ARGS__)



#define reps2(i, j, n) for(int i = 0,reps2lim=n; i < reps2lim ;++i)for(int j = 0; j < reps2lim ; ++j)

#define reps3(i, j, k, n) for(int i = 0,reps3lim=n; i < reps3lim ; ++i)for(int j = 0; j < reps3lim ; ++j)for(int k = 0; k < reps3lim ; ++k)

#define reps4(i, j, k, l, n) for(int i = 0,reps4lim=n; i < reps4lim ; ++i)for(int j = 0; j < reps4lim ; ++j)for(int k = 0; k < reps4lim ; ++k)for(int l = 0; l < reps4lim ; ++l)

#define o_reps(o1, o2, o3, o4, o5, name, ...) name

#define reps(...) o_reps(__VA_ARGS__,reps4,reps3,reps2,rep2,)(__VA_ARGS__)



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



#define fora(a, b) for(auto&& a : b)

#define forg(gi, ve) for (int gi = 0,forglim = ve.size(), f, t, c; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)

#define fort(gi, ve) for (int gi = 0, f, t, c; gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)if(t!=p)



#define form(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it).fi < r; ++it)

#define forit(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r;)



//マクロ 定数

#define k3 1010

#define k4 10101

#define k5 101010

#define k6 1010101

#define k7 10101010

const int inf = (int) 1e9 + 100;

const ll linf = (ll) 1e18 + 100;

const char infc = '{';

const string infs = "{";

const double eps = 1e-9;

const double PI = 3.1415926535897932384626433832795029L;

ll ma = numeric_limits<ll>::min();

ll mi = numeric_limits<ll>::max();

const int y4[] = {-1, 1, 0, 0};

const int x4[] = {0, 0, -1, 1};

const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};

const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};



//マクロ省略形 関数等

#define arsz(a) (sizeof(a)/sizeof(a[0]))

#define sz(a) ((int)(a).size())

#define mp make_pair

#define pb pop_back

#define pf push_front

#define eb emplace_back

#define all(a) (a).begin(),(a).end()

#define rall(a) (a).rbegin(),(a).rend()



constexpr bool ev(int a) { return !(a & 1); }

constexpr bool od(int a) { return (a & 1); }

//@拡張系 こう出来るべきというもの



//埋め込み 存在を意識せずに機能を増やされているもの

//@formatter:on

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); }};

}

//@formatter:off

//stream まとめ

istream &operator>>(istream &iss, P &a) {iss >> a.first >> a.second;return iss;}template<typename T> istream &operator>>(istream &iss, vector<T> &vec) {for (T &x: vec) iss >> x;return iss;}template<class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {os << p.fi << " " << p.se << endl;return os;}ostream &operator<<(ostream &os, T p) {    os << p.f << " " << p.s << " " << p.t;    return os;}ostream &operator<<(ostream &os, F p) {    os << p.a << " " << p.b << " " << p.c << " " << p.d;    return os;}template<typename T> ostream &operator<<(ostream &os, vector <T> &vec) {    for (int i = 0; i < vec.size(); ++i)os << vec[i] << (i + 1 == vec.size() ? "" : " ");    return os;}template<typename T> ostream &operator<<(ostream &os, vector <vector<T>> &vec) {    for (int i = 0; i < vec.size(); ++i) {        for (int j = 0; j < vec[0].size(); ++j) {            os << vec[i][j];        }        os << endl;    }    return os;}template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {    for (auto &&v:m) os << v;    return os;}

template<typename 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> int 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> int rfind_if2(vector<T> &v, F f) {    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }    return -1;}

template<class T> bool contains(string &s, const T &v) { return s.find(v) != string::npos; }

template<typename T> bool contains(vector<T> &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); }

template<typename T, typename F> bool contains_if2(vector<T> &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); }

template<typename T, typename F> int count_if2(T &v, F f) { return f(v); }

template<typename T, typename F> int count_if2(vector<T> &vec, F f) {    int ret = 0;    fora(v, vec)ret += count_if2(v, f);    return ret;}

template<typename T, typename F> void for_each2(T &v, F f) { f(v); }

template<typename T, typename F> void for_each2(vector<T> &vec, F f) { fora(v, vec)for_each2(v, f); }

template<typename W> int count_od(vector<W> &a) {return count_if2(a,[](int v){return v&1 ;});}

template<typename W> int count_ev(vector<W> &a) {return count_if2(a,[](int v){return !(v&1) ;});}

#define all_of(a,right) all_of2(a,lam(right))

#define any_of(a,right) any_of2(a,lam(right))

#define none_of(a,right) none_of2(a,lam(right))

#define find_if(a,right) find_if2(a,lam(right))

#define rfind_if(a,right) rfind_if2(a,lam(right))

#define contains_if(a,right) contains_if2(a,lam(right))

#define count_if(a, right) count_if2(a,lam(right))

#define for_each(a, right) do{fora(v,a){v right;}}while(0)





template<class T, class U> void replace(vector<T> &a, T key, U v) { replace(a.begin(), a.end(), key, v); }

void replace(str &a, char key, str v) { if (v == "")a.erase(remove(all(a), key), a.end()); }

void replace(str &a, char key, char v) { replace(all(a), key, v); }

//keyと同じかどうか01で置き換える

template<class T, class U> void replace(vector<T> &a, U k) { 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;int kn = sz(key);std::string::size_type Pos(a.find(key));int l = 0;while (Pos != std::string::npos) {t << a.substr(l, Pos - l);t << v;l = Pos + kn;Pos = a.find(key, Pos + kn);}t << a.substr(l, sz(a) - l);a = t.str();}

template<class T> bool includes(vector<T> &a, vector<T> &b) {vi c = a;vi d = b;sort(all(c));sort(all(d));return includes(all(c), all(d));}

template<class T> bool is_permutation(vector<T> &a, vector<T> &b) { return is_permutation(all(a), all(b)); }

template<class T> bool next_permutation(vector<T> &a) { return next_permutation(all(a)); }

void iota(vector<int> &ve, int s, int n) {ve.resize(n);iota(all(ve), s);}

vi iota(int s, int len) {vi ve(len);iota(all(ve), s);return ve;}

template<class A, class B> auto vtop(vector<A> &a, vector<B> &b) {    assert(sz(a) == sz(b));    /*stringを0で初期化できない  */  vector<pair<A, B>> res;    rep(i, sz(a))res.eb(a[i], b[i]);return res;}

template<class A, class B> void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {    a.resize(sz(p)), b.resize(sz(p));    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;}

template<class A, class B, class C> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {    assert(sz(a) == sz(b) && sz(b) == sz(c));    vector<T2<A, B, C>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i]);    return res;}

template<class A, class B, class C, class D> auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {    assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));    vector<F2<A, B, C, D>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i], d[i]);    return res;}

enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };

enum tcomparator {    fisiti, fisitd, fisdti, fisdtd, fdsiti, fdsitd, fdsdti, fdsdtd,    fitisi, fitisd, fitdsi, fitdsd, fdtisi, fdtisd, fdtdsi, fdtdsd,    sifiti, sifitd, sifdti, sifdtd, sdfiti, sdfitd, sdfdti, sdfdtd,    sitifi, sitifd, sitdfi, sitdfd, sdtifi, sdtifd, sdtdfi, sdfdfd,    tifisi, tifisd, tifdsi, tifdsd, tdfisi, tdfisd, tdfdsi, tdfdsd,    tisifi, tisifd, tisdfi, tisdfd, tdsifi, tdsifd, tdsdfi, tdsdfd};

template<class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {    typedef pair<A, B> U;    if (type == fisi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });};template<class U> void sort(vector<U> &a, pcomparator type) {    if (type == fisi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });};template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D> > &a, pcomparator type) {    typedef F2<A, B, C, D> U;    if (type == fisi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });};template<class U> void sort(vector<U> &a, tcomparator type) {    if (type == 0) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });    else if (type == 1) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });    else if (type == 2) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });    else if (type == 3) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });    else if (type == 4) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });    else if (type == 5) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });    else if (type == 6) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });    else if (type == 7) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });    else if (type == 8) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });    else if (type == 9) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });    else if (type == 10) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });    else if (type == 11) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });    else if (type == 12) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });    else if (type == 13) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });    else if (type == 14) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });    else if (type == 15) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });    else if (type == 16) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });    else if (type == 17) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });    else if (type == 18) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });    else if (type == 19) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });    else if (type == 20) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });    else if (type == 21) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });    else if (type == 22) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });    else if (type == 23) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });    else if (type == 24) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });    else if (type == 25) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });    else if (type == 26) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });    else if (type == 27) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });    else if (type == 28) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });    else if (type == 29) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });    else if (type == 30) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });    else if (type == 31) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });    else if (type == 32) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == 33) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == 34) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == 35) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == 36) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == 37) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == 38) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == 39) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == 40) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == 41) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == 42) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == 43) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });    else if (type == 44) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == 45) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == 46) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == 47) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });}template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {    typedef F2<A, B, C, D> U;    if (type == 0) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });    else if (type == 1) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });    else if (type == 2) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });    else if (type == 3) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });    else if (type == 4) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });    else if (type == 5) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });    else if (type == 6) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });    else if (type == 7) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });    else if (type == 8) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });    else if (type == 9) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });    else if (type == 10) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });    else if (type == 11) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });    else if (type == 12) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });    else if (type == 13) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });    else if (type == 14) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });    else if (type == 15) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });    else if (type == 16) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });    else if (type == 17) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });    else if (type == 18) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });    else if (type == 19) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });    else if (type == 20) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });    else if (type == 21) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });    else if (type == 22) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });    else if (type == 23) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });    else if (type == 24) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });    else if (type == 25) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });    else if (type == 26) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });    else if (type == 27) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });    else if (type == 28) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });    else if (type == 29) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });    else if (type == 30) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });    else if (type == 31) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });    else if (type == 32) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == 33) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == 34) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == 35) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == 36) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == 37) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == 38) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == 39) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == 40) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == 41) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == 42) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == 43) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });    else if (type == 44) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == 45) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == 46) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == 47) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });}



void sort(string &a) { sort(all(a)); }

template<class T> void sort(vector<T> &a) { sort(all(a)); }

//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), [&](int x, int y) { return f(a[x]) < f(a[y]); });    return ind;}template<class T> vi rsorti(vector<T> &a) {    auto b = a;    vi ind = iota(0, sz(a));    rsortp(b, ind);    return ind;}template<class T, class F> vi rsorti(vector<T> &a, F f) {    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(a[x]) > f(a[y]); });    return ind;}template<class A, class B, class F> vi sortpi(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(c[x]) < f(c[y]); });    return ind;}template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b, pcomparator f) {    vi ind = iota(0, sz(a));    auto c = a;    auto d = b;    sortt(c, d, ind, f);    return ind;}template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fisi); };template<class A, class B, class F> vi rsortpi(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(c[x]) > f(c[y]); });    return ind;}template<class A, class B> vi rsortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fdsd); };template<class A, class B, class C, class F> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(d[x]) < f(d[y]); });    return ind;}template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, pcomparator f) {    vi ind = iota(0, sz(a));    auto d = vtof(a, b, c, ind);    sort(d, f);    rep(i, sz(a))ind[i] = d[i].d;    return ind;}template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c) {    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) {        if (a[x] == a[y]) {            if (b[x] == b[y])return c[x] < c[y];            else return b[x] < b[y];        } else {            return a[x] < a[y];        }    });    return ind;}template<class A, class B, class C, class F> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(d[x]) > f(d[y]); });    return ind;}template<class A, class B, class C> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c) {    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) {        if (a[x] == a[y]) {            if (b[x] == b[y])return c[x] > c[y];            else return b[x] > b[y];        } else {            return a[x] > a[y];        }    });    return ind;}

template<class T> void sort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; ++i)sort(a[i]); }

template<class T> void rsort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; ++i)rsort(a[i]); }



template<typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) { rep(i, N)a[i] = v; }template<typename A, size_t N, size_t O, typename T> void fill(A (&a)[N][O], const T &v) { rep(i, N)rep(j, O)a[i][j] = v; }template<typename A, size_t N, size_t O, size_t P, typename T> void fill(A (&a)[N][O][P], const T &v) { rep(i, N)rep(j, O)rep(k, P)a[i][j][k] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, typename T> void fill(A (&a)[N][O][P][Q], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)a[i][j][k][l] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, typename T> void fill(A (&a)[N][O][P][Q][R], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)a[i][j][k][l][m] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S, typename T> void fill(A (&a)[N][O][P][Q][R][S], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)a[i][j][k][l][m][n] = v; }

template<typename 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<class T,class U>void fill(vector<T> &a,U val,vi& ind) {fora(v,ind)a[v]=val;}

template<typename A, size_t N> A sum(A (&a)[N]) {    A res = 0;    rep(i, N)res += a[i];    return res;}template<typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {    A res = 0;    rep(i, N)rep(j, O)res += a[i][j];    return res;}template<typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)res += a[i][j][k];    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> A sum(A (&a)[N][O][P][Q]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)res += a[i][j][k][l];    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A sum(A (&a)[N][O][P][Q][R]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)res += a[i][j][k][l][m];    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A sum(A (&a)[N][O][P][Q][R][S]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)res += a[i][j][k][l][m][n];    return res;}

//@汎用便利関数 入力

int in() {int ret;cin >> ret;return ret;}

string sin() {string ret;cin >> ret;return ret;}

template<class T>  void in(T &head) { cin >> head; }template<class T, class... U>  void in(T &head, U &... tail) {cin >> head;in(tail...);}



#define o_din(o1, o2, o3, o4, name, ...) name

#define din1(a) int a;cin>>a

#define din2(a, b) int a,b;cin>>a>> b

#define din3(a, b, c) int a,b,c;cin>>a>>b>>c

#define din4(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d

#define din(...) o_din(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__)



#define o_dind(o1, o2, o3, o4, name, ...) name

#define din1d(a) din1(a);a--

#define din2d(a, b) din2(a,b);a--,b--

#define din3d(a, b, c) din3(a,b,c);a--,b--,c--

#define din4d(a, b, c, d) din4(a,b,c,d);a--,b--,c--,d--

#define dind(...) o_dind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)





#define o_out(o1, o2, o3, o4, name, ...) name

#define out1(a) cout<<a<<endl

#define out2(a, b) cout<<a<<" "<< b<<endl

#define out3(a, b, c) cout<<a<<" "<<b<<" "<<c<<endl

#define out4(a, b, c, d) cout<<a<<" "<<b<<" "<<c<<" "<<d<<endl

#define out(...) o_out((__VA_ARGS__),out4,out3,out2,out1)((__VA_ARGS__))



template<class T>void outv(vector<T> &a,int W){    rep(i,W){        cout <<a[i]<< " ";    }    cout << "" << endl;}template<class T>void outv(vector<vector<T> > &a,int H,int W){    rep(h, H) {        rep(w, W) {            cout << a[h][w] << " ";        }       cout << "" << endl;    }}



template<class T> void outl(vector<T> &a) { fora(v, a)cout << v << endl; }

template<class T> void na(vector<T> &a, int n) {a.resize(n);rep(i, n)cin >> a[i];}

#define dna(a, n) vi a(n); rep(dnai,n) cin >> a[dnai];

template<class T> void nao(vector<T> &a, int n) {    a.resize(n + 1);    a[0] = 0;    rep(i, n)cin >> a[i + 1];}

template<class T> void naod(vector<T> &a, int 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, int n) {    a.resize(n);    rep(i, n)cin >> a[i], a[i]--;}

template<class T, class U> void na2(vector<T> &a, vector<U> &b, int n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i];}

#define dna2(a, b, n) vi a(n),b(n);rep(dna2i, n)cin >> a[dna2i] >> b[dna2i];

template<class T, class U> void nao2(vector<T> &a, vector<U> &b, int n) {    a.resize(n + 1);    b.resize(n + 1);    a[0] = b[0] = 0;    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, int n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i], a[i]--, b[i]--;}

template<class T, class U, class W> void na3(vector<T> &a, vector<U> &b, vector<W> &c, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i];}

#define dna3(a, b, c, n) vi a(n),b(n),c(n);   rep(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, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;}

#define dna3d(a, b, c, n) vi a(n),b(n),c(n);  rep(dna3di, n){cin >> a[dna3di] >> b[dna3di] >> c[dna3di];a[dna3di]--,b[dna3di]--,c[dna3di]--;}

#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 debugName(VariableName) # VariableName



#define deb1(x)  debugName(x)<<" = "<<x

#define deb2(x, ...) deb1(x) <<", "<< deb1(__VA_ARGS__)

#define deb3(x, ...) deb1(x) <<", "<< deb2(__VA_ARGS__)

#define deb4(x, ...) deb1(x) <<", "<< deb3(__VA_ARGS__)

#define deb5(x, ...) deb1(x) <<", "<< deb4(__VA_ARGS__)

#define deb6(x, ...) deb1(x) <<", "<< deb5(__VA_ARGS__)

#define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)

#define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)

#define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)

#define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)



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



#ifdef _DEBUG

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

#else

#define deb(...) ;

#endif



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



//@formatter:on

//よく使うクラス、構造体

struct unionfind {

    vector<int> par;

    vector<int> siz;

    vector<int> es;

    int n, trees;//連結グループの数(親の種類)

    unionfind(int n) : n(n), trees(n) {

        par.resize(n);

        siz.resize(n);

        es.resize(n);

        for (int i = 0; i < n; i++) {

            par[i] = i;

            siz[i] = 1;

        }

    }

    int root(int x) { if (par[x] == x) { return x; } else { return par[x] = root(par[x]); }}

    void unite(int x, int y) {

        x = root(x);

        y = root(y);

        es[x]++;

        if (x == y) return;

        if (siz[x] > siz[y]) swap(x, y);

        trees--;

        par[x] = y;

        siz[y] += siz[x];

        es[y] += es[x];

    }

    bool same(int x, int y) { return root(x) == root(y); }

    int size(int x) { return siz[root(x)]; }

    int esize(int x) { return es[root(x)]; }

    //つながりを無向グラフと見なし、xが閉路に含まれるか判定

    bool close(int x) { return esize(x) >= size(x); }

    /*@formatter:off*/V<vi> sets() {vvi(res, trees);        umap<int, vi> map;        rep(i, n) map[root(i)].push_back(i);        int i = 0;        for (auto &&p:map) {            int r = p.fi;            res[i].push_back(r);            for (auto &&v:p.se) {                if (r == v)continue;                res[i].push_back(v);            }            ++i;        }        return res;    }

};



using bint =__int128;

using u32 = unsigned;

using u64 = unsigned long long;

using u128 = __uint128_t;



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

//__int128 toi128(string &s) {    __int128 ret = 0;    for (int i = 0; i < s.length(); ++i)        if ('0' <= s[i] && s[i] <= '9')            ret = 10 * ret + s[i] - '0';    return ret;}





//エラー

void ole() {

#ifdef _DEBUG

    debugline("ole");exit(0);

#endif

    string a = "a";rep(i, 30)a += a;rep(i, 1 << 17)cout << a << endl;cout << "OLE 出力長制限超過" << endl;exit(0);

}

void re() { assert(0 == 1);exit(0); }

void tle() { while (inf)cout << inf << endl; }



//便利関数



//テスト用

char ranc() {return (char) ('a' + rand() % 26);}

int rand(int min, int max) {    assert(min <= max);    if (min >= 0 && max >= 0) {        return rand() % (max + 1 - min) + min;    } else if (max < 0) {        return -rand(-max, -min);    } else {        if (rand() % 2) {            return rand(0, max);        } else {            return -rand(0, -min);        }    }}

vi ranv(int n, int min, int max) {    vi v(n);    rep(i, n)v[i] = rand(min, max);    return v;}

str ransu(int n) {    str s;    rep(i, n)s += (char) rand('A', 'Z');    return s;}

str ransl(int n) {    str s;    rep(i, n)s += (char) rand('a', 'z');    return s;}

//単調増加

vi ranvinc(int n, int min, int max) {    vi v(n);    bool bad = 1;    while (bad) {        bad = 0;        v.resize(n);        rep(i, n) {            if (i && min > max - v[i - 1]) {                bad = 1;                break;            }            if (i)v[i] = v[i - 1] + rand(min, max - v[i - 1]);            else v[i] = rand(min, max);        }    }    return v;}

//便利 汎用

void ranvlr(int n, int min, int max, vi &l, vi &r) {    l.resize(n);    r.resize(n);    rep(i, n) {        l[i] = rand(min, max);        r[i] = l[i] + rand(0, max - l[i]);    }}

vp run_length(vi &a) {    vp ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) {        if (ret.back().fi == a[i]) {            ret.back().se++;        } else {            ret.eb(a[i], 1);        }    }    return ret;}

vector<pair<char, int>> run_length(string &a) {    vector<pair<char, int>> ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) {        if (ret.back().fi == a[i]) {            ret.back().se++;        } else {            ret.eb(a[i], 1);        }    }    return ret;}

template<class F>  int mgr(int ok, int ng, F f) {if (ok < ng)while (ng - ok > 1) { int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid; } else while (ok - ng > 1) { int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid; }return ok;}

//strを整数として比較

string smax(str &a, str b) {    if (sz(a) < sz(b)) {        return b;    } else if (sz(a) > sz(b)) {        return a;    } else {        rep(i, sz(a)) {            if (a[i] < b[i]) {                return b;            } else if (a[i] > b[i])return a;        }    }    return a;}

//strを整数として比較

string smin(str &a, str b) {    if (sz(a) < sz(b)) {        return a;    } else if (sz(a) > sz(b)) {        return b;    } else {        rep(i, sz(a)) {            if (a[i] < b[i]) {                return a;            } else if (a[i] > b[i])return b;        }    }    return a;}

template<typename W, typename T> int 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);}





template<typename W, typename T> int count2(W &a, const T k) { return a == k; }

template<typename W, typename T> int count2(vector<W> &a, const T k) {int ret = 0;fora(v, a)ret +=count(v, k);return ret;}

template<typename W, typename T> int count(vector<W> &a, const T k) {int ret = 0;fora(v, a)ret +=count2(v, k);return ret;}

int count(str& a, str k) {int ret = 0, len = k.length();auto pos = a.find(k);while (pos != string::npos)pos = a.find(k, pos + len), ++ret;return ret;}

vi count(str& a){    vi cou(26);    char c='a';    if('A' <= a[0] && a[0]<='Z')c='A';    rep(i,sz(a))++cou[a[i]-c];    return cou;}

#define couif count_if

//algorythm





inline ll rev(ll a) {ll res = 0;while (a) {res *= 10;res += a % 10;a /= 10;}return res;}

template<class T> void rev(vector<T> &a) {reverse(all(a));}

template<class 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 inline rev(string &a) {reverse(all(a));}

constexpr int p10[] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000ll, 100000000000ll, 1000000000000ll, 10000000000000ll, 100000000000000ll, 1000000000000000ll, 10000000000000000ll, 100000000000000000ll, 1000000000000000000ll};



int get(int a, int keta) { return (a / (int) pow(10, keta)) % 10; }

int keta(ll v) { if (v < p10[9]) { if (v < p10[4]) { if (v < p10[2]) { if (v < p10[1]) return 1; else return 2; } else { if (v < p10[3]) return 3; else return 4; }} else { if (v < p10[7]) { if (v < p10[5]) return 5; else if (v < p10[6])return 6; else return 7; } else { if (v < p10[8])return 8; else return 9; }}} else { if (v < p10[13]) { if (v < p10[11]) { if (v < p10[10]) return 10; else return 11; } else { if (v < p10[12]) return 12; else return 13; }} else { if (v < p10[15]) { if (v < p10[14]) return 14; else if (v < p10[15])return 15; else return 16; } else { if (v < p10[17])return 17; else return 18; }}}}

int dsum(int v) {int ret = 0;for (; v; v /= 10)ret += v % 10;return ret;}



struct sint {

    int v;

    sint(int v) : v(v) {}

    operator int() { return v; }

    //下からi番目

    int operator[](int i) { return (v / p10[i]) % 10; }

    int back(int i) {        return operator[](i);    }

    //上からi番目

    int top(int i) {        int len = keta(v);        return operator[](len - 1 - i);    }

    //先頭からi番目にセット

    int settop(int i, int k) {        int len = keta(v);        return set(len - 1 - i, k);    }

    int set(int i, int k) {        if (i < 0)return settop(abs(i) - 1, k);        return v += p10[i] * (k - (v / p10[i]) % 10);    }

    int add(int i, int k = 1) {        return v += p10[i] * k;    }

    int addtop(int i, int k = 1) {        return v += p10[keta(v) - i - 1] * k;    }

    int dec(int i, int k = 1) {        return v -= p10[i] * k;    }

    int dectop(int i, int k = 1) {        return v -= p10[keta(v) - i - 1] * k;    }

#define op(t, o)template<class T>inline t operator o(T r){return v o r;}

    op(int, +=);op(int, -=);op(int, *=);op(int, /=);op(int, %=);

    op(int, +);op(int, -);op(int, *);op(int, /);op(int, %);

    op(bool, ==);op(bool, !=);op(bool, <);op(bool, <=);op(bool, >);op(bool, >=);

#undef op

    template<class T> inline int operator<<=(T r) { return v *= p10[r]; }

    template<class T> inline int operator<<(T r) { return v * p10[r]; }

    template<class T> inline int operator>>=(T r) { return v /= p10[r]; }

    template<class T> inline int operator>>(T r) { return v / p10[r]; }

};

int mask10(int v) { return p10[v] - 1; }

//変換系

template<class T> auto keys(T a) {vector<decltype((a.begin())->fi)> res;for (auto &&k :a)res.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> inline bool chma(T &a, const U &b) {    if (a < b) {        a = b;        return true;    }    return false;}

template<class U> inline bool chma(const U &b) { return chma(ma, b); }

template<class T, class U> inline bool chmi(T &a, const U &b) {    if (b < a) {        a = b;        return true;    }    return false;}

template<class U> inline bool chmi(const U &b) { return chmi(mi, b); }

template<class T> inline T min(T a, signed b) { return a < b ? a : b; }

template<class T> inline T max(T a, signed b) { return a < b ? b : a; }

template<class T> inline T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }

template<class T> inline T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }

template<class T> inline T min(vector<T> a) { return *min_element(all(a)); }

template<class T> inline T mini(vector<T> a) { return min_element(all(a)) - a.begin(); }

template<class T> inline T min(vector<T> a, int n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T min(vector<T> a, int s, int n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a) { return *max_element(all(a)); }

template<class T> inline T maxi(vector<T> a) { return max_element(all(a)) - a.begin(); }

template<class T> inline T max(vector<T> a, int n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a, int s, int n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<typename A, size_t N> inline A max(A (&a)[N]) {    A res = a[0];    rep(i, N)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O> inline A max(A (&a)[N][O]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> inline A max(A (&a)[N][O][P]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> inline A max(A (&a)[N][O][P][Q], const T &v) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> inline A max(A (&a)[N][O][P][Q][R]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> inline A max(A (&a)[N][O][P][Q][R][S]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N> inline A min(A (&a)[N]) {    A res = a[0];    rep(i, N)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O> inline A min(A (&a)[N][O]) {    A res = min(a[0]);    rep(i, N)res = min(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> inline A min(A (&a)[N][O][P]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> inline A min(A (&a)[N][O][P][Q], const T &v) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> inline A min(A (&a)[N][O][P][Q][R]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> inline A min(A (&a)[N][O][P][Q][R][S]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}

template<class T> T sum(vector<T> &v, int s = 0, int t = inf) {    T ret = 0;    rep(i, s, min(sz(v), t))ret += v[i];    return ret;}

template<class T> T sum(vector<vector<T> > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T> T sum(vector<vector<vector<T> > > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T> T sum(vector<vector<vector<vector<T> > > > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T> T sum(vector<vector<vector<vector<vector<T> > > > > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T>  auto sum(priority_queue<T, vector<T>, greater<T> > &r) {auto q = r;T ret = 0;while (sz(q)) {ret += q.top();q.pop();}return ret;}

template<class T>  auto sum(priority_queue<T> &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}

//template<class T, class U, class... W> inline 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;}

void clear(PQ &q) { q=PQ(); }

template<class T>  void clear(queue<T> &q) { while (q.size())q.pop(); }

template<class T>  T *negarr(int size) {    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));    return body + size;}

template<class T>  T *negarr2(int h, int w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (int i = 1; i <= 2 * h + 1; ++i) {        dummy1[i] = dummy1[i - 1] + 2 * w + 1;    }    double **a = dummy1 + h;    return a;}

//imoは0-indexed

//ruiは1-indexed

template<class T> vector<T> imo(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];    return ret;}

//kと同じものの数

template<class T, class U>  vi imocou(vector<T> &a, U k) {    vector<T> ret = a;    rep(i, sz(ret) - 1)ret[i + 1] = ret[i] + (a[i] == k);    return    ret;}

template<class T>  vector<T> imox(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}

//漸化的に最小を持つ

template<class T>  vector<T> imi(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}

template<class T> struct ruiC {    const vector<T> rui;    ruiC(vector<T> &ru) : rui(ru) {}    T operator()(int l, int r) {        assert(l <= r);        return rui[r] - rui[l];    }    T operator[](int i) { return rui[i]; }    T back() { return rui.back(); }    int size() { return rui.size(); }};

template<class T> struct rruic {    const T *rrui;    rruic(T *ru) : rrui(ru) {}    inline T operator()(int l, int r) {        assert(l >= r);        return rrui[r] - rrui[l];    }    inline T operator[](int i) { return rrui[i]; }};

template<class T>  vector<T> ruiv(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i];    return ret;}

template<class T>  ruiC<T> ruic(vector<T> &a) {    vector<T> ret = ruiv(a);    return ruiC<T>(ret);}

vector<int> ruiv(string &a) {    if (sz(a) == 0)return vi(1);    int dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;    vector<int> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec;    return ret;}

ruiC<int> ruic(string &a) {    vector<int> ret = ruiv(a);    return ruiC<int>(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<int> ruic(T &a, U k) {    vi ret = ruiv(a, k);    return ruiC<int>(ret);}

//xor

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>  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) {    int n = sz(a);    vector<T> ret(n + 1);    rep(i, 1, n) {        ret[i] = a[i - 1];        chmi(ret[i + 1], ret[i]);    }    return ret;}

//template<class T> T *rrui(vector<T> &a) {

//右から左にかけての半開区間 (-1 n-1]

template<class T> rruic<T> rrui(vector<T> &a) {    int len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    rer(i, len - 1)res[i - 1] = res[i] + a[i];    return rruic<T>(res);}

//掛け算

template<class T> T *rruim(vector<T> &a) {    int len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    res[len - 1] = 1;    rer(i, len - 1)res[i - 1] = res[i] * a[i];    return res;}

template<class T, class U> void inc(T &a, U v = 1) { a += v; }

template<class T, class U> void inc(vector<T> &a, U v = 1) { for (auto &u:a)inc(u, v); }

template<class T, class U> void dec(T &a, U v = 1) { a -= v; }

template<class T, class U> void dec(vector<T> &a, U v = 1) { for (auto &u :a)dec(u, v); }

template<class U> void dec(string &a, U v = 1) { for (auto &u :a)dec(u, v); }

template<class T> void dec(vector<T> &a) { for (auto &u :a)dec(u, 1); }

bool ins(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; }

bool ins(int l, int v, int r) { return l <= v && v < r; }

template<class T>bool ins(vector<T>& a,int i,int j=0){return ins(0,i,sz(a)) &&ins(0,j,sz(a));}

ll u(ll a) { return a < 0 ? 0 : a; }

template<class T> vector<T> u(const vector<T> &a) {vector<T> ret = a;fora(v, ret)v = u(v);return ret;}

#define MIN(a) numeric_limits<a>::min()

#define MAX(a) numeric_limits<a>::max()

template<class F>ll goldd(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(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;}

//@formatter:on

//lld でなくdoubleではどうか

template<class F> flo golddd(flo left, flo right, F calc) {

    flo GRATIO = 1.6180339887498948482045868343656;

    flo lm = left + ((right - left) / (GRATIO + 1.0));

    flo rm = lm + ((right - lm) / (GRATIO + 1.0));

    flo fl = calc(lm);

    flo fr = calc(rm);

    rep(i, 200) {

        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;

}



//@formatter:off

template<class T> T min(vector<vector<T >> &a) {T res = MAX(T);rep(i, a.size())chmi(res, *min_element(all(a[i])));return res;}

template<class T> T max(vector<vector<T >> &a) {T res = MIN(T);rep(i, a.size())chma(res, *max_element(all(a[i])));return res;}



constexpr bool bget(ll m, int keta) {return (m >> keta) & 1;}

int bget(ll m, int keta, int sinsuu) {m /= (ll) pow(sinsuu, keta);return m % sinsuu;}

ll bit(int n) { return (1LL << (n)); }

ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); }

int mask(int n) { return (1ll << n) - 1; }

#define bcou __builtin_popcountll

//最下位ビット

int lbit(int n) {return n & -n;}

//最上位ビット

int hbit(int n) {n |= (n >> 1);n |= (n >> 2);n |= (n >> 4);n |= (n >> 8);n |= (n >> 16);n |= (n >> 32);return n - (n >> 1);}

int hbitk(int n) {int k = 0;rer(i, 5) {int a = k + (1ll << i);int b = 1ll << a;if (b <= n)k += 1ll << i;}return k;}

//初期化は0を渡す

ll nextComb(ll &mask, int n, int r) {    if (!mask)return mask = (1LL << r) - 1;    ll x = mask & -mask; /*最下位の1*/    ll y = mask + x; /*連続した下の1を繰り上がらせる*/    ll res = ((mask & ~y) / x >> 1) | y;    if (bget(res, n))return mask = 0;    else return mask = res;}

//n桁以下でビットがr個立っているもののvectorを返す

vi bitCombList(int n, int r) {vi res;int m = 0;while (nextComb(m, n, r)) {res.push_back(m);}return res;}

char itoal(int i) {    return 'a' + i;}

char itoaL(int i) {    return 'A' + i;}

int altoi(char c) {    if ('A' <= c && c <= 'Z')return c - 'A';    return c - 'a';}

int ctoi(char c) { return c - '0'; }

char itoc(int i) { return i + '0'; }

int vtoi(vi &v) {    int res = 0;    if (sz(v) > 18) {        debugline("vtoi");        deb(sz(v));        ole();    }    rep(i, sz(v)) {        res *= 10;        res += v[i];    }    return res;}

vi itov(int i) {    vi res;    while (i) {        res.push_back(i % 10);        i /= 10;    }    rev(res);    return res;}

vi stov(string &a) {    int n = sz(a);    vi ret(n);    rep(i, n) {        ret[i] = a[i] - '0';    }    return ret;}

//基準を満たさないものは0になる

vi stov(string &a,char one) {    int n=sz(a);    vi ret (n);    rep(i,n)ret[i]=a[i]==one;    return ret;}

vector<vector<int>> ctoi(vector<vector<char>> s, char c) {    int n = sz(s), m = sz(s[0]);    vector<vector<int>> res(n, vector<int>(m));    rep(i, n)rep(j, m)res[i][j] = s[i][j] == c;    return res;}

#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );

//[i] := i番として圧縮されたものを返す

vi compress(vi &a) {    vi b;    int len = a.size();    for (int i = 0; i < len; ++i) {        b.push_back(a[i]);    }    sort(b);    unique(b);    for (int i = 0; i < len; ++i) {        a[i] = lower_bound(all(b), a[i]) - b.begin();    }    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vi &a, umap<int, int> &map) {    vi b;    int len = a.size();    for (int i = 0; i < len; ++i) {        b.push_back(a[i]);    }    sort(b);    unique(b);    for (int i = 0; i < len; ++i) {        int v = a[i];        a[i] = lower_bound(all(b), a[i]) - b.begin();        map[v] = a[i];    }    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vi &a, vi &r) {    vi b;    int len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    sort(b);    unique(b);    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vi &a, vi &r, vi &s) {    vi b;    int len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    fora(v, s)b.push_back(v);    sort(b);    unique(b);    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    for (int i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(V<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();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vector<vector<vi >> &a) {    vi b;    fora(vvv, a)fora(vv, vvv)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

void compress(int a[], int len) {    vi b;    for (int i = 0; i < len; ++i) {        b.push_back(a[i]);    }    sort(b);    unique(b);    for (int i = 0; i < len; ++i) {        a[i] = lower_bound(all(b), a[i]) - b.begin();    }}

//要素が見つからなかったときに困る

#define binarySearch(a, v) (binary_search(all(a),v))

#define lowerIndex(a, v) (lower_bound(all(a),v)-a.begin())

#define lowerBound(a, v) (*lower_bound(all(a),v))

#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())

#define upperBound(a, v) (*upper_bound(all(a),v))

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



//便利 数学 math

int mod(int a, int m) { return (a % m + m) % m; }

int pow(int a) { return a * a; };

ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }



ll comi(int n, int r) {    assert(n < 100);    static vvi(pas, 100, 100);    if (pas[0][0])return pas[n][r];    pas[0][0] = 1;    rep(i, 1, 100) {        pas[i][0] = 1;        rep(j, 1, i + 1)pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];    }    return pas[n][r];}

double comd(int n, int r) {    static vd fac;    if (sz(fac) < n + 1) {        if (sz(fac) == 0)fac.push_back(1);        rep(i, sz(fac) - 1, n) {            fac.push_back(fac.back() * (i + 1));        }    }    if (n < r || n <= 0)return 0;    return fac[n] / fac[n - r] / fac[r];}

int gcd(int a, int b) {    while (b) a %= b, swap(a, b);    return abs(a);}

int gcd(vi b) {    ll res = b[0];    rep(i, 1, sz(b))res = gcd(b[i], res);    return res;}

ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }

ll lcm(vi a) {    int res = a[0];    rep(i, 1, sz(a))res = lcm(a[i], res);    return res;}

ll ceil(ll a, ll b) {    if (b == 0) {        debugline("ceil");        deb(a, b);        ole();        return -1;    } else if (a < 0) {        return 0;    } else {return (a + b - 1) / b;}}

int lower_remi__bx_a(int kei, int rem, int x) {    if (rem >= x) return 0;    return (x - rem + kei - 1) / kei;}

int lower_remv__bx_a(int kei, int rem, int x) {    if (rem >= x) return rem;    return (x - rem + kei - 1) / kei * kei + rem;}

int upper_remi__bx_a(int kei, int rem, int x) {    if (rem > x) return 0;    return (x - rem + kei) / kei;}

int upper_remv__bx_a(int kei, int rem, int x) {    if (rem > x) return rem;    return (x - rem + kei) / kei * kei + rem;}



//v * v >= aとなる最小のvを返す

ll sqrt(ll a) {if (a < 0) {debugline("sqrt");deb(a);ole();}ll res = (ll) std::sqrt(a);while (res * res < a)++res;return res;}

double log(double e, double x) { return log(x) / log(e); }

ll sig(ll t) { return (1 + t) * t / 2; }

ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }





//幾何 Pをcomplexとして扱う

template<class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }

dou atan2(pd a) { return atan2(a.se, a.fi); }

dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }

dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }

//bを中心とするabcのtheta aからcにかけて時計回り

dou angle(pd a, pd b, pd c) {    dou ax = a.fi - b.fi;    dou ay = a.se - b.se;    dou cx = c.fi - b.fi;    dou cy = c.se - b.se;    double ret = atan2(cy, cx) - atan2(ay, ax);    if (ret < 0) ret += 2 * PI;    return ret;}

dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }

dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }



//機能拡張

template<class T, class U> void operator+=(queue<T> &a, U v) { a.push(v); }

template<class T, class U> void operator+=(deque<T> &a, U v) { a.push_back(v); }

template<class T> 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> set<T> &operator+=(set<T> &a, U v) {    a.insert(v);    return a;}

template<class T, class U> set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {    a.insert(v);    return a;}

template<class T, class U> vector<T> &operator+=(vector<T> &a, U v) {    a.push_back(v);    return a;}

template<class T, class U> vector<T> operator+(const vector <T> &a, U v) {vector<T> ret = a;ret += v;return ret;}

template<class T, class U> vector<T> operator+(U v, const vector <T> &a) {vector<T> ret = a;ret.insert(ret.begin(), v);return ret;}

template<class T> vector<T> operator+(vector<T> a, vector <T> b) {vector<T> ret;ret = a;fora(v, b)ret += v;return ret;}

template<class T> vector<T> &operator+=(vector<T> &a, vector <T> &b) {fora(v, b)a += v;return a;}

template<class T> vector<T> &operator-=(vector<T> &a, vector <T> &b) {if (sz(a) != sz(b)) {debugline("vector<T> operator-=");deb(a);deb(b);exit(0);}rep(i, sz(a))a[i] -= b[i];return a;}

template<class T> vector<T> operator-(vector<T> &a, vector <T> &b) {if (sz(a) != sz(b)) {debugline("vector<T> operator-");deb(a);deb(b);ole();}vector<T> res(sz(a));rep(i, sz(a))res[i] = a[i] - b[i];return res;}

template<class T,class U> vector<T> operator*(vector<T> &a, U b) {vector<T> ret;fora(v,a)ret+=v*b;return ret;}

template<class T,class U> vector<T> operator/(vector<T> &a, U b) {vector<T> ret;fora(v,a)ret+=v/b;return ret;}

template<class T,class U> vector<T> operator*=(vector<T> &a, U b) {fora(v,a)v*=b;return a;}

template<class T,class U> vector<T> operator/=(vector<T> &a, U b) {fora(v,a)v/=b;return a;}

template<typename T> void erase(vector<T> &v, unsigned int i) {v.erase(v.begin()+ i);}

template<typename T> void erase(vector<T> &v, unsigned int s,unsigned int e) {v.erase(v.begin()+ s, v.begin()+ e);}

template<class T, class U> void erase(map<T, U> &m, int okl, int ngr) {m.erase(m.lower_bound(okl), m.lower_bound(ngr));}

template<class T> void erase(set<T> &m, int okl, int ngr) {m.erase(m.lower_bound(okl), m.lower_bound(ngr));}

template<typename T> void erasen(vector<T> &v, unsigned int s,unsigned int n) {v.erase(v.begin()+ s, v.begin()+ s + n);}

template<typename T, typename U> void insert(vector<T> &v, unsigned int i,U t) {v.insert(v.begin()+ i, t);}

template<typename T, typename U> void push_front(vector<T> &v, U t) {v.insert(v.begin(), t);}

template<typename T, typename U> void insert(vector<T> &v, unsigned int i,vector<T> list) {for (auto &&va:list)v.insert(v.begin()+ i++, va);}

template<typename T> void insert(set<T> &v, vector<T> list) {for (auto &&va :list)v.insert(va);}

vector<string> split(const string a, const char deli) {    string b = a + deli;    int l = 0, r = 0, n = b.size();    vector<string> res;    rep(i, n) {        if (b[i] == deli) {            r = i;            if (l < r)res.push_back(b.substr(l, r - l));            l = i + 1;        }    }    return res;}

vector<string> split(const string a, const string deli) {    vector<string> res;    int kn = sz(deli);    std::string::size_type Pos(a.find(deli));    int l = 0;    while (Pos != std::string::npos) {        if (Pos - l)res.push_back(a.substr(l, Pos - l));        l = Pos + kn;        Pos = a.find(deli, Pos + kn);    }    if (sz(a) - l)res.push_back(a.substr(l, sz(a) - l));    return res;}

void yn(bool a) {if (a)cout << "yes" << endl;    else cout << "no" << endl;}

void Yn(bool a) {if (a)cout << "Yes" << endl;    else cout << "No" << endl;}

void YN(bool a) {if (a)cout << "YES" << endl;    else cout << "NO" << endl;}

void fyn(bool a) {if (a)cout << "yes" << endl;    else cout << "no" << endl;    exit(0);}

void fYn(bool a) {if (a)cout << "Yes" << endl;    else cout << "No" << endl;    exit(0);}

void fYN(bool a) {if (a)cout << "YES" << endl;    else cout << "NO" << endl;    exit(0);}

void Possible(bool a) {if (a)cout << "Possible" << endl;    else cout << "Impossible" << endl;    exit(0);}

void POSSIBLE(bool a) {if (a)cout << "POSSIBLE" << endl;    else cout << "IMPOSSIBLE" << endl;    exit(0);}

//@起動時

struct initon {

    initon() {

        cin.tie(0);

        ios::sync_with_stdio(false);

        cout.setf(ios::fixed);

        cout.precision(16);

        srand((unsigned) clock() + (unsigned) time(NULL));

    };

} initonv;



//@formatter:on

//gra mint pr

int n, m, k, d, H, W, x, y, z, q;

int cou;

vi t, a, b, c;

//vvi (s, 0, 0);

vvc (ba, 0, 0);

vp p;

str s;





void solve() {

    in(n);

    vi x, y;

    vc d;

    na3(x, y, d, n);

    auto cal = [&](dou v) {

        dou mix = linf;

        dou miy = linf;

        dou max = -linf;

        dou may = -linf;

        rep(i, n) {

            dou nx = x[i], ny = y[i];

            if (d[i] == 'R')nx += v;

            elf (d[i] == 'L')nx -= v;

            elf (d[i] == 'U')ny += v;

            elf (d[i] == 'D')ny -= v;

            chmi(mix, nx);

            chma(max, nx);

            chmi(miy, ny);

            chma(may, ny);

        }

        return (max - mix) * (may - miy);

    };

    dou i = golddd(0.0, 1e18, cal);

    dou best = cal(i);

    chmi(best, cal(golddd(0.0, 1e8 + 300, cal)));

    chmi(best, cal(golddd(0.0, 1e8 / 3, cal)));

    chmi(best, cal(golddd(1e8 / 3 - 100, 1e8 / 3 * 2 + 300, cal)));

    chmi(best, cal(golddd(1e8 / 3 * 2 - 300, 1e8 + 300, cal)));

    dou l = 0;

    rep(i, 11) {

        dou r = l + 1e8 / 10 + 100;

        chmi(best, cal(golddd(l, r, cal)));

        l += 1e8 / 10 - 10;

    }

    cout << best << endl;

}

int my(int n, vi &a) {

    return 0;

}

int sister(int n, vi &a) {

    int ret = 0;

    return ret;

}



signed main() {

    solve();



#define arg n,a

#ifdef _DEBUG

    bool bad = 0;

    for (int i = 0, ok = 1; i < k5 && ok; ++i) {

        int n = rand(1, 8);

        vi a = ranv(n, 1, 10);

        int myres = my(arg);

        int res = sister(arg);

        ok = myres == res;

        if (!ok) {

            out(arg);

            cerr << "正解 : " << res << endl;

            cerr << "自分  : " << myres << endl;

            bad = 1;

            break;

        }

    }

    if (!bad) {

//        cout << "完璧 : solveを書き直そう" << endl;

//        cout << "     : そして、solve()を呼び出すのだ" << endl;

//        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;

    }

#endif

    return 0;

};




// Optimized solution to the same problem (faster):
//#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;

#define int long long

#define ll long long



//@formatter:off

#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(int k) {        auto it = s.begin();        for (int i = 0; i < k; i++)it++;        return it;    }    auto order_by_key(int v) {        auto it = s.begin();        int i=0;        for (;it != s.end() && *it <v ; i++)it++;        return i;    }};

#define pbds(T) my_pbds_tree<T>

#else

#define unordered_map __gnu_pbds::gp_hash_table

//find_by_order(k) k番目のイテレーター

//order_of_key(k)  k以上が前から何番目か

#define pbds(U) __gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update>

#endif

struct xorshift {    static uint64_t splitmix64(uint64_t x) {        x += 0x9e3779b97f4a7c15;        x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;        x = (x ^ (x >> 27)) * 0x94d049bb133111eb;        return x ^ (x >> 31);    }    size_t operator()(uint64_t x) const {        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();        return splitmix64(x + FIXED_RANDOM);    }    size_t operator()(std::pair<ll, ll> x) const {        int v=((x.first) << 32) | x.second;        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();        return splitmix64(v + FIXED_RANDOM);    }};

template<class U, class L> void operator+=(__gnu_pbds::tree<U, __gnu_pbds::null_type, less<U>, __gnu_pbds::rb_tree_tag, __gnu_pbds::tree_order_statistics_node_update> &s, L v) { s.insert(v); }

//衝突対策

#define ws wszzzz



template<class A, class B, class C>struct T2 {A f;B s;C t;T2() { f = 0, s = 0, t = 0; }T2(A f, B s, C t) : f(f), s(s), t(t) {}bool operator<(const T2 &r) const {        return f != r.f ? f < r.f : s != r.s ? s < r.s : t < r.t;        /*return f != r.f ? f > r.f : s != r.s ?n s > r.s : t > r.t; 大きい順 */   }    bool operator>(const T2 &r) const {        return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t;        /*return f != r.f ? f > r.f : s != r.s ? s > r.s : t > r.t; 小さい順 */   }    bool operator==(const T2 &r) const {        return f == r.f && s == r.s && t == r.t;    }    bool operator!=(const T2 &r) const {        return f != r.f || s != r.s || t != r.t;    }};

template<class A, class B, class C, class D> struct F2 {    A a;    B b;    C c;    D d;    F2() { a = 0, b = 0, c = 0, d = 0; }    F2(A a, B b, C c, D d) : a(a), b(b), c(c), d(d) {}    bool operator<(const F2 &r) const {        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c < r.c : d < r.d;    /*    return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c > r.c : d > r.d;*/    }    bool operator>(const F2 &r) const {        return a != r.a ? a > r.a : b != r.b ? b > r.b : c != r.c ? c > r.c : d > r.d;/*        return a != r.a ? a < r.a : b != r.b ? b < r.b : c != r.c ? c < r.c : d < r.d;*/    }    bool operator==(const F2 &r) const {        return a == r.a && b == r.b && c == r.c && d == r.d;    }    bool operator!=(const F2 &r) const {        return a != r.a || b != r.b || c != r.c || d != r.d;    }    int operator[](int i) {        assert(i < 4);        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;    }};

typedef T2<int, int, int> T;

typedef F2<int, int, int, int> F;

T mt(int a, int b, int c) {return T(a, b, c);}



//@マクロ省略系 型,構造

#define double long double

#define ull unsigned long long

using dou = double;

using itn = int;

using str = string;

using bo= bool;

#define au auto

using P = pair<int, int>;

using pd =pair<dou, dou>;

#define fi first

#define se second

#define beg begin

#define rbeg rbegin

#define con continue

#define bre break

#define brk break

#define is ==

#define elf else if

#define wh while



#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) [&](int& p){return p right;}



//マクロ省略系 コンテナ

using vi = vector<int>;

using vb = vector<bool>;

using vs = vector<string>;

using vd = vector<double>;

using vc = vector<char>;

using vp = vector<P>;

using vt = vector<T>;



#define V vector

#define o_vvt(o1, o2, o3, o4, name, ...) name

#define vvt0(t) V<V<t>>

#define vvt1(t,a) V<V<t>>a

#define vvt2(t,a, b) V<V<t>>a(b)

#define vvt3(t,a, b, c) V<V<t>> a(b,V<t>(c))

#define vvt4(t,a, b, c, d) V<V<t>> a(b,V<t>(c,d))



#define vvi(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(int,__VA_ARGS__)

#define vvb(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(bool,__VA_ARGS__)

#define vvs(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(string,__VA_ARGS__)

#define vvd(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(double,__VA_ARGS__)

#define vvc(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(char,__VA_ARGS__)

#define vvp(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(P,__VA_ARGS__)



template<typename T> vector<T> make_v(size_t a) { return vector<T>(a); }

template<typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));}

#define vni(name, ...) auto name = make_v<int>(__VA_ARGS__)

#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)

#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)

#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)

#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)

#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)



#define PQ priority_queue<int, vector<int>, greater<int> >

#define tos to_string

using mapi = map<int, int>;

using mapp = map<P, int>;

using mapd = map<dou, int>;

using mapc = map<char, int>;

using maps = map<str, int>;

using seti = set<int>;

using setd = set<dou>;

using setc = set<char>;

using sets = set<str>;

using qui = queue<int>;

#define bset bitset

#define uset unordered_set

#define useti unordered_set<int,int,xorshift>

#define mset multiset

#define umap unordered_map

#define umapi unordered_map<int,int,xorshift>

#define umapp unordered_map<P,int,xorshift>

#define mmap multimap



template<class T> struct pq {    priority_queue<T, vector<T>, greater<T> > q;/*小さい順*/    T su = 0;    void clear() {q = priority_queue<T, vector<T>, greater<T> >();su = 0;}    void operator+=(T v) {su += v;q.push(v);}    T sum() {return su;}    T top() {return q.top();}    void pop() {su -= q.top();q.pop();}    T poll() {T ret = q.top();su -= ret;q.pop();return ret;}    int size() {return q.size();}};

template<class T> struct pqg {    priority_queue<T> q;/*大きい順*/    T su = 0;    void clear() {q = priority_queue<T>();su = 0;}    void operator+=(T v) {su += v;q.push(v);}    T sum() {return su;}    T top() {return q.top();}    void pop() {su -= q.top();q.pop();}    T poll() {T ret = q.top();su -= ret;q.pop();return ret;}    int size() {return q.size();}};

#define pqi pq<int>

#define pqgi pqg<int>

//マクロ 繰り返し

#define o_rep(o1, o2, o3, o4, name, ...) name

# define rep1(n) for(int rep1i = 0,rep1lim=n; rep1i < rep1lim ; ++rep1i)

# define rep2(i, n) for(int i = 0,rep2lim=n; i < rep2lim ; ++i)

#define rep3(i, m, n) for(int i = m,rep3lim=n; i < rep3lim ; ++i)

#define rep4(i, m, n, ad) for(int i = m,rep4lim=n; i < rep4lim ; i+= ad)

#define rep(...) o_rep(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)



#define rer2(i, n) for(int i = n; i >= 0 ; i--)

#define rer3(i, m, n) for(int i = m,rer3lim=n; i >= rer3lim ; i--)

#define rer4(i, m, n, dec) for(int i = m,rer4lim=n; i >= rer4lim ; i-=dec)

#define rer(...) o_rep(__VA_ARGS__,rer4,rer3,rer2,)(__VA_ARGS__)



#define reps2(i, j, n) for(int i = 0,reps2lim=n; i < reps2lim ;++i)for(int j = 0; j < reps2lim ; ++j)

#define reps3(i, j, k, n) for(int i = 0,reps3lim=n; i < reps3lim ; ++i)for(int j = 0; j < reps3lim ; ++j)for(int k = 0; k < reps3lim ; ++k)

#define reps4(i, j, k, l, n) for(int i = 0,reps4lim=n; i < reps4lim ; ++i)for(int j = 0; j < reps4lim ; ++j)for(int k = 0; k < reps4lim ; ++k)for(int l = 0; l < reps4lim ; ++l)

#define o_reps(o1, o2, o3, o4, o5, name, ...) name

#define reps(...) o_reps(__VA_ARGS__,reps4,reps3,reps2,rep2,)(__VA_ARGS__)



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



#define fora(a, b) for(auto&& a : b)

#define forg(gi, ve) for (int gi = 0,forglim = ve.size(), f, t, c; gi < forglim && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)

#define fort(gi, ve) for (int gi = 0, f, t, c; gi < ve.size() && (f = ve[gi].f, t = ve[gi].t, c = ve[gi].c, true); ++gi)if(t!=p)



#define form(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it).fi < r; ++it)

#define forit(st, l, r) for (auto &&it = st.lower_bound(l); it != st.end() && (*it) < r;)



//マクロ 定数

#define k3 1010

#define k4 10101

#define k5 101010

#define k6 1010101

#define k7 10101010

const int inf = (int) 1e9 + 100;

const ll linf = (ll) 1e18 + 100;

const char infc = '{';

const string infs = "{";

const double eps = 1e-9;

const double PI = 3.1415926535897932384626433832795029L;

ll ma = numeric_limits<ll>::min();

ll mi = numeric_limits<ll>::max();

const int y4[] = {-1, 1, 0, 0};

const int x4[] = {0, 0, -1, 1};

const int y8[] = {0, 1, 0, -1, -1, 1, 1, -1};

const int x8[] = {1, 0, -1, 0, 1, -1, 1, -1};



//マクロ省略形 関数等

#define arsz(a) (sizeof(a)/sizeof(a[0]))

#define sz(a) ((int)(a).size())

#define mp make_pair

#define pb pop_back

#define pf push_front

#define eb emplace_back

#define all(a) (a).begin(),(a).end()

#define rall(a) (a).rbegin(),(a).rend()



constexpr bool ev(int a) { return !(a & 1); }

constexpr bool od(int a) { return (a & 1); }

//@拡張系 こう出来るべきというもの



//埋め込み 存在を意識せずに機能を増やされているもの

//@formatter:on

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); }};

}

//@formatter:off

//stream まとめ

istream &operator>>(istream &iss, P &a) {iss >> a.first >> a.second;return iss;}template<typename T> istream &operator>>(istream &iss, vector<T> &vec) {for (T &x: vec) iss >> x;return iss;}template<class T, class U> ostream &operator<<(ostream &os, pair<T, U> p) {os << p.fi << " " << p.se << endl;return os;}ostream &operator<<(ostream &os, T p) {    os << p.f << " " << p.s << " " << p.t;    return os;}ostream &operator<<(ostream &os, F p) {    os << p.a << " " << p.b << " " << p.c << " " << p.d;    return os;}template<typename T> ostream &operator<<(ostream &os, vector <T> &vec) {    for (int i = 0; i < vec.size(); ++i)os << vec[i] << (i + 1 == vec.size() ? "" : " ");    return os;}template<typename T> ostream &operator<<(ostream &os, vector <vector<T>> &vec) {    for (int i = 0; i < vec.size(); ++i) {        for (int j = 0; j < vec[0].size(); ++j) {            os << vec[i][j];        }        os << endl;    }    return os;}template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {    for (auto &&v:m) os << v;    return os;}

template<typename 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> int 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> int rfind_if2(vector<T> &v, F f) {    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }    return -1;}

template<class T> bool contains(string &s, const T &v) { return s.find(v) != string::npos; }

template<typename T> bool contains(vector<T> &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); }

template<typename T, typename F> bool contains_if2(vector<T> &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); }

template<typename T, typename F> int count_if2(T &v, F f) { return f(v); }

template<typename T, typename F> int count_if2(vector<T> &vec, F f) {    int ret = 0;    fora(v, vec)ret += count_if2(v, f);    return ret;}

template<typename T, typename F> void for_each2(T &v, F f) { f(v); }

template<typename T, typename F> void for_each2(vector<T> &vec, F f) { fora(v, vec)for_each2(v, f); }

template<typename W> int count_od(vector<W> &a) {return count_if2(a,[](int v){return v&1 ;});}

template<typename W> int count_ev(vector<W> &a) {return count_if2(a,[](int v){return !(v&1) ;});}

#define all_of(a,right) all_of2(a,lam(right))

#define any_of(a,right) any_of2(a,lam(right))

#define none_of(a,right) none_of2(a,lam(right))

#define find_if(a,right) find_if2(a,lam(right))

#define rfind_if(a,right) rfind_if2(a,lam(right))

#define contains_if(a,right) contains_if2(a,lam(right))

#define count_if(a, right) count_if2(a,lam(right))

#define for_each(a, right) do{fora(v,a){v right;}}while(0)





template<class T, class U> void replace(vector<T> &a, T key, U v) { replace(a.begin(), a.end(), key, v); }

void replace(str &a, char key, str v) { if (v == "")a.erase(remove(all(a), key), a.end()); }

void replace(str &a, char key, char v) { replace(all(a), key, v); }

//keyと同じかどうか01で置き換える

template<class T, class U> void replace(vector<T> &a, U k) { 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;int kn = sz(key);std::string::size_type Pos(a.find(key));int l = 0;while (Pos != std::string::npos) {t << a.substr(l, Pos - l);t << v;l = Pos + kn;Pos = a.find(key, Pos + kn);}t << a.substr(l, sz(a) - l);a = t.str();}

template<class T> bool includes(vector<T> &a, vector<T> &b) {vi c = a;vi d = b;sort(all(c));sort(all(d));return includes(all(c), all(d));}

template<class T> bool is_permutation(vector<T> &a, vector<T> &b) { return is_permutation(all(a), all(b)); }

template<class T> bool next_permutation(vector<T> &a) { return next_permutation(all(a)); }

void iota(vector<int> &ve, int s, int n) {ve.resize(n);iota(all(ve), s);}

vi iota(int s, int len) {vi ve(len);iota(all(ve), s);return ve;}

template<class A, class B> auto vtop(vector<A> &a, vector<B> &b) {    assert(sz(a) == sz(b));    /*stringを0で初期化できない  */  vector<pair<A, B>> res;    rep(i, sz(a))res.eb(a[i], b[i]);return res;}

template<class A, class B> void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {    a.resize(sz(p)), b.resize(sz(p));    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;}

template<class A, class B, class C> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {    assert(sz(a) == sz(b) && sz(b) == sz(c));    vector<T2<A, B, C>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i]);    return res;}

template<class A, class B, class C, class D> auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {    assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));    vector<F2<A, B, C, D>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i], d[i]);    return res;}

enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };

enum tcomparator {    fisiti, fisitd, fisdti, fisdtd, fdsiti, fdsitd, fdsdti, fdsdtd,    fitisi, fitisd, fitdsi, fitdsd, fdtisi, fdtisd, fdtdsi, fdtdsd,    sifiti, sifitd, sifdti, sifdtd, sdfiti, sdfitd, sdfdti, sdfdtd,    sitifi, sitifd, sitdfi, sitdfd, sdtifi, sdtifd, sdtdfi, sdfdfd,    tifisi, tifisd, tifdsi, tifdsd, tdfisi, tdfisd, tdfdsi, tdfdsd,    tisifi, tisifd, tisdfi, tisdfd, tdsifi, tdsifd, tdsdfi, tdsdfd};

template<class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {    typedef pair<A, B> U;    if (type == fisi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });};template<class U> void sort(vector<U> &a, pcomparator type) {    if (type == fisi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });};template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D> > &a, pcomparator type) {    typedef F2<A, B, C, D> U;    if (type == fisi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });};template<class U> void sort(vector<U> &a, tcomparator type) {    if (type == 0) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });    else if (type == 1) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });    else if (type == 2) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });    else if (type == 3) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });    else if (type == 4) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });    else if (type == 5) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });    else if (type == 6) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });    else if (type == 7) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });    else if (type == 8) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });    else if (type == 9) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });    else if (type == 10) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });    else if (type == 11) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });    else if (type == 12) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });    else if (type == 13) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });    else if (type == 14) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });    else if (type == 15) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });    else if (type == 16) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });    else if (type == 17) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });    else if (type == 18) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });    else if (type == 19) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });    else if (type == 20) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });    else if (type == 21) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });    else if (type == 22) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });    else if (type == 23) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });    else if (type == 24) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });    else if (type == 25) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });    else if (type == 26) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });    else if (type == 27) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });    else if (type == 28) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });    else if (type == 29) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });    else if (type == 30) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });    else if (type == 31) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });    else if (type == 32) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == 33) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == 34) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == 35) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == 36) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == 37) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == 38) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == 39) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == 40) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == 41) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == 42) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == 43) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });    else if (type == 44) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == 45) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == 46) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == 47) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });}template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {    typedef F2<A, B, C, D> U;    if (type == 0) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });    else if (type == 1) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });    else if (type == 2) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });    else if (type == 3) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });    else if (type == 4) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });    else if (type == 5) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });    else if (type == 6) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });    else if (type == 7) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });    else if (type == 8) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });    else if (type == 9) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });    else if (type == 10) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });    else if (type == 11) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });    else if (type == 12) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });    else if (type == 13) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });    else if (type == 14) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });    else if (type == 15) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });    else if (type == 16) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });    else if (type == 17) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });    else if (type == 18) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });    else if (type == 19) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });    else if (type == 20) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });    else if (type == 21) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });    else if (type == 22) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });    else if (type == 23) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });    else if (type == 24) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });    else if (type == 25) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });    else if (type == 26) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });    else if (type == 27) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });    else if (type == 28) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });    else if (type == 29) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });    else if (type == 30) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });    else if (type == 31) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });    else if (type == 32) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == 33) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == 34) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == 35) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == 36) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == 37) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == 38) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == 39) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == 40) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == 41) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == 42) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == 43) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });    else if (type == 44) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == 45) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == 46) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == 47) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });}



void sort(string &a) { sort(all(a)); }

template<class T> void sort(vector<T> &a) { sort(all(a)); }

//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), [&](int x, int y) { return f(a[x]) < f(a[y]); });    return ind;}template<class T> vi rsorti(vector<T> &a) {    auto b = a;    vi ind = iota(0, sz(a));    rsortp(b, ind);    return ind;}template<class T, class F> vi rsorti(vector<T> &a, F f) {    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(a[x]) > f(a[y]); });    return ind;}template<class A, class B, class F> vi sortpi(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(c[x]) < f(c[y]); });    return ind;}template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b, pcomparator f) {    vi ind = iota(0, sz(a));    auto c = a;    auto d = b;    sortt(c, d, ind, f);    return ind;}template<class A, class B> vi sortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fisi); };template<class A, class B, class F> vi rsortpi(vector<A> &a, vector<B> &b, F f) {    auto c = vtop(a, b);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(c[x]) > f(c[y]); });    return ind;}template<class A, class B> vi rsortpi(vector<A> &a, vector<B> &b) { return sortpi(a, b, fdsd); };template<class A, class B, class C, class F> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(d[x]) < f(d[y]); });    return ind;}template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c, pcomparator f) {    vi ind = iota(0, sz(a));    auto d = vtof(a, b, c, ind);    sort(d, f);    rep(i, sz(a))ind[i] = d[i].d;    return ind;}template<class A, class B, class C> vi sortti(vector<A> &a, vector<B> &b, vector<C> &c) {    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) {        if (a[x] == a[y]) {            if (b[x] == b[y])return c[x] < c[y];            else return b[x] < b[y];        } else {            return a[x] < a[y];        }    });    return ind;}template<class A, class B, class C, class F> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c, F f) {    auto d = vtot(a, b, c);    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) { return f(d[x]) > f(d[y]); });    return ind;}template<class A, class B, class C> vi rsortti(vector<A> &a, vector<B> &b, vector<C> &c) {    vi ind = iota(0, sz(a));    sort(all(ind), [&](int x, int y) {        if (a[x] == a[y]) {            if (b[x] == b[y])return c[x] > c[y];            else return b[x] > b[y];        } else {            return a[x] > a[y];        }    });    return ind;}

template<class T> void sort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; ++i)sort(a[i]); }

template<class T> void rsort2(vector<vector<T >> &a) { for (int i = 0, n = a.size(); i < n; ++i)rsort(a[i]); }



template<typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) { rep(i, N)a[i] = v; }template<typename A, size_t N, size_t O, typename T> void fill(A (&a)[N][O], const T &v) { rep(i, N)rep(j, O)a[i][j] = v; }template<typename A, size_t N, size_t O, size_t P, typename T> void fill(A (&a)[N][O][P], const T &v) { rep(i, N)rep(j, O)rep(k, P)a[i][j][k] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, typename T> void fill(A (&a)[N][O][P][Q], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)a[i][j][k][l] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, typename T> void fill(A (&a)[N][O][P][Q][R], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)a[i][j][k][l][m] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S, typename T> void fill(A (&a)[N][O][P][Q][R][S], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)a[i][j][k][l][m][n] = v; }

template<typename 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<class T,class U>void fill(vector<T> &a,U val,vi& ind) {fora(v,ind)a[v]=val;}

template<typename A, size_t N> A sum(A (&a)[N]) {    A res = 0;    rep(i, N)res += a[i];    return res;}template<typename A, size_t N, size_t O> A sum(A (&a)[N][O]) {    A res = 0;    rep(i, N)rep(j, O)res += a[i][j];    return res;}template<typename A, size_t N, size_t O, size_t P> A sum(A (&a)[N][O][P]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)res += a[i][j][k];    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> A sum(A (&a)[N][O][P][Q]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)res += a[i][j][k][l];    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> A sum(A (&a)[N][O][P][Q][R]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)res += a[i][j][k][l][m];    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> A sum(A (&a)[N][O][P][Q][R][S]) {    A res = 0;    rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)res += a[i][j][k][l][m][n];    return res;}

//@汎用便利関数 入力

int in() {int ret;cin >> ret;return ret;}

string sin() {string ret;cin >> ret;return ret;}

template<class T>  void in(T &head) { cin >> head; }template<class T, class... U>  void in(T &head, U &... tail) {cin >> head;in(tail...);}



#define o_din(o1, o2, o3, o4, name, ...) name

#define din1(a) int a;cin>>a

#define din2(a, b) int a,b;cin>>a>> b

#define din3(a, b, c) int a,b,c;cin>>a>>b>>c

#define din4(a, b, c, d) int a,b,c,d;cin>>a>>b>>c>>d

#define din(...) o_din(__VA_ARGS__,din4,din3,din2 ,din1)(__VA_ARGS__)



#define o_dind(o1, o2, o3, o4, name, ...) name

#define din1d(a) din1(a);a--

#define din2d(a, b) din2(a,b);a--,b--

#define din3d(a, b, c) din3(a,b,c);a--,b--,c--

#define din4d(a, b, c, d) din4(a,b,c,d);a--,b--,c--,d--

#define dind(...) o_dind(__VA_ARGS__,din4d,din3d,din2d ,din1d)(__VA_ARGS__)





#define o_out(o1, o2, o3, o4, name, ...) name

#define out1(a) cout<<a<<endl

#define out2(a, b) cout<<a<<" "<< b<<endl

#define out3(a, b, c) cout<<a<<" "<<b<<" "<<c<<endl

#define out4(a, b, c, d) cout<<a<<" "<<b<<" "<<c<<" "<<d<<endl

#define out(...) o_out((__VA_ARGS__),out4,out3,out2,out1)((__VA_ARGS__))



template<class T>void outv(vector<T> &a,int W){    rep(i,W){        cout <<a[i]<< " ";    }    cout << "" << endl;}template<class T>void outv(vector<vector<T> > &a,int H,int W){    rep(h, H) {        rep(w, W) {            cout << a[h][w] << " ";        }       cout << "" << endl;    }}



template<class T> void outl(vector<T> &a) { fora(v, a)cout << v << endl; }

template<class T> void na(vector<T> &a, int n) {a.resize(n);rep(i, n)cin >> a[i];}

#define dna(a, n) vi a(n); rep(dnai,n) cin >> a[dnai];

template<class T> void nao(vector<T> &a, int n) {    a.resize(n + 1);    a[0] = 0;    rep(i, n)cin >> a[i + 1];}

template<class T> void naod(vector<T> &a, int 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, int n) {    a.resize(n);    rep(i, n)cin >> a[i], a[i]--;}

template<class T, class U> void na2(vector<T> &a, vector<U> &b, int n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i];}

#define dna2(a, b, n) vi a(n),b(n);rep(dna2i, n)cin >> a[dna2i] >> b[dna2i];

template<class T, class U> void nao2(vector<T> &a, vector<U> &b, int n) {    a.resize(n + 1);    b.resize(n + 1);    a[0] = b[0] = 0;    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, int n) {    a.resize(n);    b.resize(n);    rep(i, n)cin >> a[i] >> b[i], a[i]--, b[i]--;}

template<class T, class U, class W> void na3(vector<T> &a, vector<U> &b, vector<W> &c, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i];}

#define dna3(a, b, c, n) vi a(n),b(n),c(n);   rep(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, int n) {    a.resize(n);    b.resize(n);    c.resize(n);    rep(i, n)cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;}

#define dna3d(a, b, c, n) vi a(n),b(n),c(n);  rep(dna3di, n){cin >> a[dna3di] >> b[dna3di] >> c[dna3di];a[dna3di]--,b[dna3di]--,c[dna3di]--;}

#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 debugName(VariableName) # VariableName



#define deb1(x)  debugName(x)<<" = "<<x

#define deb2(x, ...) deb1(x) <<", "<< deb1(__VA_ARGS__)

#define deb3(x, ...) deb1(x) <<", "<< deb2(__VA_ARGS__)

#define deb4(x, ...) deb1(x) <<", "<< deb3(__VA_ARGS__)

#define deb5(x, ...) deb1(x) <<", "<< deb4(__VA_ARGS__)

#define deb6(x, ...) deb1(x) <<", "<< deb5(__VA_ARGS__)

#define deb7(x, ...) deb1(x) <<", "<< deb6(__VA_ARGS__)

#define deb8(x, ...) deb1(x) <<", "<< deb7(__VA_ARGS__)

#define deb9(x, ...) deb1(x) <<", "<< deb8(__VA_ARGS__)

#define deb10(x, ...) deb1(x) <<", "<< deb9(__VA_ARGS__)



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



#ifdef _DEBUG

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

#else

#define deb(...) ;

#endif



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



//@formatter:on

//よく使うクラス、構造体

struct unionfind {

    vector<int> par;

    vector<int> siz;

    vector<int> es;

    int n, trees;//連結グループの数(親の種類)

    unionfind(int n) : n(n), trees(n) {

        par.resize(n);

        siz.resize(n);

        es.resize(n);

        for (int i = 0; i < n; i++) {

            par[i] = i;

            siz[i] = 1;

        }

    }

    int root(int x) { if (par[x] == x) { return x; } else { return par[x] = root(par[x]); }}

    void unite(int x, int y) {

        x = root(x);

        y = root(y);

        es[x]++;

        if (x == y) return;

        if (siz[x] > siz[y]) swap(x, y);

        trees--;

        par[x] = y;

        siz[y] += siz[x];

        es[y] += es[x];

    }

    bool same(int x, int y) { return root(x) == root(y); }

    int size(int x) { return siz[root(x)]; }

    int esize(int x) { return es[root(x)]; }

    //つながりを無向グラフと見なし、xが閉路に含まれるか判定

    bool close(int x) { return esize(x) >= size(x); }

    /*@formatter:off*/V<vi> sets() {vvi(res, trees);        umap<int, vi> map;        rep(i, n) map[root(i)].push_back(i);        int i = 0;        for (auto &&p:map) {            int r = p.fi;            res[i].push_back(r);            for (auto &&v:p.se) {                if (r == v)continue;                res[i].push_back(v);            }            ++i;        }        return res;    }

};



using bint =__int128;

using u32 = unsigned;

using u64 = unsigned long long;

using u128 = __uint128_t;



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

//__int128 toi128(string &s) {    __int128 ret = 0;    for (int i = 0; i < s.length(); ++i)        if ('0' <= s[i] && s[i] <= '9')            ret = 10 * ret + s[i] - '0';    return ret;}





//エラー

void ole() {

#ifdef _DEBUG

    debugline("ole");exit(0);

#endif

    string a = "a";rep(i, 30)a += a;rep(i, 1 << 17)cout << a << endl;cout << "OLE 出力長制限超過" << endl;exit(0);

}

void re() { assert(0 == 1);exit(0); }

void tle() { while (inf)cout << inf << endl; }



//便利関数



//テスト用

char ranc() {return (char) ('a' + rand() % 26);}

int rand(int min, int max) {    assert(min <= max);    if (min >= 0 && max >= 0) {        return rand() % (max + 1 - min) + min;    } else if (max < 0) {        return -rand(-max, -min);    } else {        if (rand() % 2) {            return rand(0, max);        } else {            return -rand(0, -min);        }    }}

vi ranv(int n, int min, int max) {    vi v(n);    rep(i, n)v[i] = rand(min, max);    return v;}

str ransu(int n) {    str s;    rep(i, n)s += (char) rand('A', 'Z');    return s;}

str ransl(int n) {    str s;    rep(i, n)s += (char) rand('a', 'z');    return s;}

//単調増加

vi ranvinc(int n, int min, int max) {    vi v(n);    bool bad = 1;    while (bad) {        bad = 0;        v.resize(n);        rep(i, n) {            if (i && min > max - v[i - 1]) {                bad = 1;                break;            }            if (i)v[i] = v[i - 1] + rand(min, max - v[i - 1]);            else v[i] = rand(min, max);        }    }    return v;}

//便利 汎用

void ranvlr(int n, int min, int max, vi &l, vi &r) {    l.resize(n);    r.resize(n);    rep(i, n) {        l[i] = rand(min, max);        r[i] = l[i] + rand(0, max - l[i]);    }}

vp run_length(vi &a) {    vp ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) {        if (ret.back().fi == a[i]) {            ret.back().se++;        } else {            ret.eb(a[i], 1);        }    }    return ret;}

vector<pair<char, int>> run_length(string &a) {    vector<pair<char, int>> ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) {        if (ret.back().fi == a[i]) {            ret.back().se++;        } else {            ret.eb(a[i], 1);        }    }    return ret;}

template<class F>  int mgr(int ok, int ng, F f) {if (ok < ng)while (ng - ok > 1) { int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid; } else while (ok - ng > 1) { int mid = (ok + ng) / 2; if (f(mid))ok = mid; else ng = mid; }return ok;}

//strを整数として比較

string smax(str &a, str b) {    if (sz(a) < sz(b)) {        return b;    } else if (sz(a) > sz(b)) {        return a;    } else {        rep(i, sz(a)) {            if (a[i] < b[i]) {                return b;            } else if (a[i] > b[i])return a;        }    }    return a;}

//strを整数として比較

string smin(str &a, str b) {    if (sz(a) < sz(b)) {        return a;    } else if (sz(a) > sz(b)) {        return b;    } else {        rep(i, sz(a)) {            if (a[i] < b[i]) {                return a;            } else if (a[i] > b[i])return b;        }    }    return a;}

template<typename W, typename T> int 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);}





template<typename W, typename T> int count2(W &a, const T k) { return a == k; }

template<typename W, typename T> int count2(vector<W> &a, const T k) {int ret = 0;fora(v, a)ret +=count(v, k);return ret;}

template<typename W, typename T> int count(vector<W> &a, const T k) {int ret = 0;fora(v, a)ret +=count2(v, k);return ret;}

int count(str& a, str k) {int ret = 0, len = k.length();auto pos = a.find(k);while (pos != string::npos)pos = a.find(k, pos + len), ++ret;return ret;}

vi count(str& a){    vi cou(26);    char c='a';    if('A' <= a[0] && a[0]<='Z')c='A';    rep(i,sz(a))++cou[a[i]-c];    return cou;}

#define couif count_if

//algorythm





inline ll rev(ll a) {ll res = 0;while (a) {res *= 10;res += a % 10;a /= 10;}return res;}

template<class T> void rev(vector<T> &a) {reverse(all(a));}

template<class 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 inline rev(string &a) {reverse(all(a));}

constexpr int p10[] = {1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000ll, 100000000000ll, 1000000000000ll, 10000000000000ll, 100000000000000ll, 1000000000000000ll, 10000000000000000ll, 100000000000000000ll, 1000000000000000000ll};



int get(int a, int keta) { return (a / (int) pow(10, keta)) % 10; }

int keta(ll v) { if (v < p10[9]) { if (v < p10[4]) { if (v < p10[2]) { if (v < p10[1]) return 1; else return 2; } else { if (v < p10[3]) return 3; else return 4; }} else { if (v < p10[7]) { if (v < p10[5]) return 5; else if (v < p10[6])return 6; else return 7; } else { if (v < p10[8])return 8; else return 9; }}} else { if (v < p10[13]) { if (v < p10[11]) { if (v < p10[10]) return 10; else return 11; } else { if (v < p10[12]) return 12; else return 13; }} else { if (v < p10[15]) { if (v < p10[14]) return 14; else if (v < p10[15])return 15; else return 16; } else { if (v < p10[17])return 17; else return 18; }}}}

int dsum(int v) {int ret = 0;for (; v; v /= 10)ret += v % 10;return ret;}



struct sint {

    int v;

    sint(int v) : v(v) {}

    operator int() { return v; }

    //下からi番目

    int operator[](int i) { return (v / p10[i]) % 10; }

    int back(int i) {        return operator[](i);    }

    //上からi番目

    int top(int i) {        int len = keta(v);        return operator[](len - 1 - i);    }

    //先頭からi番目にセット

    int settop(int i, int k) {        int len = keta(v);        return set(len - 1 - i, k);    }

    int set(int i, int k) {        if (i < 0)return settop(abs(i) - 1, k);        return v += p10[i] * (k - (v / p10[i]) % 10);    }

    int add(int i, int k = 1) {        return v += p10[i] * k;    }

    int addtop(int i, int k = 1) {        return v += p10[keta(v) - i - 1] * k;    }

    int dec(int i, int k = 1) {        return v -= p10[i] * k;    }

    int dectop(int i, int k = 1) {        return v -= p10[keta(v) - i - 1] * k;    }

#define op(t, o)template<class T>inline t operator o(T r){return v o r;}

    op(int, +=);op(int, -=);op(int, *=);op(int, /=);op(int, %=);

    op(int, +);op(int, -);op(int, *);op(int, /);op(int, %);

    op(bool, ==);op(bool, !=);op(bool, <);op(bool, <=);op(bool, >);op(bool, >=);

#undef op

    template<class T> inline int operator<<=(T r) { return v *= p10[r]; }

    template<class T> inline int operator<<(T r) { return v * p10[r]; }

    template<class T> inline int operator>>=(T r) { return v /= p10[r]; }

    template<class T> inline int operator>>(T r) { return v / p10[r]; }

};

int mask10(int v) { return p10[v] - 1; }

//変換系

template<class T> auto keys(T a) {vector<decltype((a.begin())->fi)> res;for (auto &&k :a)res.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> inline bool chma(T &a, const U &b) {    if (a < b) {        a = b;        return true;    }    return false;}

template<class U> inline bool chma(const U &b) { return chma(ma, b); }

template<class T, class U> inline bool chmi(T &a, const U &b) {    if (b < a) {        a = b;        return true;    }    return false;}

template<class U> inline bool chmi(const U &b) { return chmi(mi, b); }

template<class T> inline T min(T a, signed b) { return a < b ? a : b; }

template<class T> inline T max(T a, signed b) { return a < b ? b : a; }

template<class T> inline T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }

template<class T> inline T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }

template<class T> inline T min(vector<T> a) { return *min_element(all(a)); }

template<class T> inline T mini(vector<T> a) { return min_element(all(a)) - a.begin(); }

template<class T> inline T min(vector<T> a, int n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T min(vector<T> a, int s, int n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a) { return *max_element(all(a)); }

template<class T> inline T maxi(vector<T> a) { return max_element(all(a)) - a.begin(); }

template<class T> inline T max(vector<T> a, int n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }

template<class T> inline T max(vector<T> a, int s, int n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }

template<typename A, size_t N> inline A max(A (&a)[N]) {    A res = a[0];    rep(i, N)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O> inline A max(A (&a)[N][O]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> inline A max(A (&a)[N][O][P]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> inline A max(A (&a)[N][O][P][Q], const T &v) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> inline A max(A (&a)[N][O][P][Q][R]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> inline A max(A (&a)[N][O][P][Q][R][S]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N> inline A min(A (&a)[N]) {    A res = a[0];    rep(i, N)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O> inline A min(A (&a)[N][O]) {    A res = min(a[0]);    rep(i, N)res = min(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P> inline A min(A (&a)[N][O][P]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q> inline A min(A (&a)[N][O][P][Q], const T &v) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R> inline A min(A (&a)[N][O][P][Q][R]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S> inline A min(A (&a)[N][O][P][Q][R][S]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}

template<class T> T sum(vector<T> &v, int s = 0, int t = inf) {    T ret = 0;    rep(i, s, min(sz(v), t))ret += v[i];    return ret;}

template<class T> T sum(vector<vector<T> > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T> T sum(vector<vector<vector<T> > > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T> T sum(vector<vector<vector<vector<T> > > > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T> T sum(vector<vector<vector<vector<vector<T> > > > > &v) {T ret = 0;rep(i, sz(v))ret += sum(v[i]);return ret;}

template<class T>  auto sum(priority_queue<T, vector<T>, greater<T> > &r) {auto q = r;T ret = 0;while (sz(q)) {ret += q.top();q.pop();}return ret;}

template<class T>  auto sum(priority_queue<T> &r) {    auto q = r;    T ret = 0;    while (sz(q)) {        ret += q.top();        q.pop();    }    return ret;}

//template<class T, class U, class... W> inline 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;}

void clear(PQ &q) { q=PQ(); }

template<class T>  void clear(queue<T> &q) { while (q.size())q.pop(); }

template<class T>  T *negarr(int size) {    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));    return body + size;}

template<class T>  T *negarr2(int h, int w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (int i = 1; i <= 2 * h + 1; ++i) {        dummy1[i] = dummy1[i - 1] + 2 * w + 1;    }    double **a = dummy1 + h;    return a;}

//imoは0-indexed

//ruiは1-indexed

template<class T> vector<T> imo(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];    return ret;}

//kと同じものの数

template<class T, class U>  vi imocou(vector<T> &a, U k) {    vector<T> ret = a;    rep(i, sz(ret) - 1)ret[i + 1] = ret[i] + (a[i] == k);    return    ret;}

template<class T>  vector<T> imox(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}

//漸化的に最小を持つ

template<class T>  vector<T> imi(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}

template<class T> struct ruiC {    const vector<T> rui;    ruiC(vector<T> &ru) : rui(ru) {}    T operator()(int l, int r) {        assert(l <= r);        return rui[r] - rui[l];    }    T operator[](int i) { return rui[i]; }    T back() { return rui.back(); }    int size() { return rui.size(); }};

template<class T> struct rruic {    const T *rrui;    rruic(T *ru) : rrui(ru) {}    inline T operator()(int l, int r) {        assert(l >= r);        return rrui[r] - rrui[l];    }    inline T operator[](int i) { return rrui[i]; }};

template<class T>  vector<T> ruiv(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i];    return ret;}

template<class T>  ruiC<T> ruic(vector<T> &a) {    vector<T> ret = ruiv(a);    return ruiC<T>(ret);}

vector<int> ruiv(string &a) {    if (sz(a) == 0)return vi(1);    int dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;    vector<int> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec;    return ret;}

ruiC<int> ruic(string &a) {    vector<int> ret = ruiv(a);    return ruiC<int>(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<int> ruic(T &a, U k) {    vi ret = ruiv(a, k);    return ruiC<int>(ret);}

//xor

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>  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) {    int n = sz(a);    vector<T> ret(n + 1);    rep(i, 1, n) {        ret[i] = a[i - 1];        chmi(ret[i + 1], ret[i]);    }    return ret;}

//template<class T> T *rrui(vector<T> &a) {

//右から左にかけての半開区間 (-1 n-1]

template<class T> rruic<T> rrui(vector<T> &a) {    int len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    rer(i, len - 1)res[i - 1] = res[i] + a[i];    return rruic<T>(res);}

//掛け算

template<class T> T *rruim(vector<T> &a) {    int len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    res[len - 1] = 1;    rer(i, len - 1)res[i - 1] = res[i] * a[i];    return res;}

template<class T, class U> void inc(T &a, U v = 1) { a += v; }

template<class T, class U> void inc(vector<T> &a, U v = 1) { for (auto &u:a)inc(u, v); }

template<class T, class U> void dec(T &a, U v = 1) { a -= v; }

template<class T, class U> void dec(vector<T> &a, U v = 1) { for (auto &u :a)dec(u, v); }

template<class U> void dec(string &a, U v = 1) { for (auto &u :a)dec(u, v); }

template<class T> void dec(vector<T> &a) { for (auto &u :a)dec(u, 1); }

bool ins(int h, int w, int H, int W) { return h >= 0 && w >= 0 && h < H && w < W; }

bool ins(int l, int v, int r) { return l <= v && v < r; }

template<class T>bool ins(vector<T>& a,int i,int j=0){return ins(0,i,sz(a)) &&ins(0,j,sz(a));}

ll u(ll a) { return a < 0 ? 0 : a; }

template<class T> vector<T> u(const vector<T> &a) {vector<T> ret = a;fora(v, ret)v = u(v);return ret;}

#define MIN(a) numeric_limits<a>::min()

#define MAX(a) numeric_limits<a>::max()

template<class F>ll goldd(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(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;}

//@formatter:on

//lld でなくdoubleではどうか

template<class F> dou golddd(dou left, dou right, F calc) {

    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

    int k = 141;

    while (--k) {

        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 golddd_s(dou l, dou r, F calc, int time = 2000) {

    clock_t lim = time;

    dou mini = 0, minv = MAX(dou);

    //区間をk分割する

    rep(k, 1, inf) {

        clock_t s = clock();

        dou haba = (r - l) / k;

        dou nl = l;

        dou nr = l + haba;

        rep(i, k) {

            dou ni = golddd(nl, nr, calc);

            if (chmi(minv, calc(ni))) mini = ni;

            nl = nr;

            nr = nl + haba;

        }

        clock_t now = clock();

        clock_t dif = now - s;

        if (now + dif * 2 >= lim)break;

    }

    return mini;

}





//@formatter:off

template<class T> T min(vector<vector<T >> &a) {T res = MAX(T);rep(i, a.size())chmi(res, *min_element(all(a[i])));return res;}

template<class T> T max(vector<vector<T >> &a) {T res = MIN(T);rep(i, a.size())chma(res, *max_element(all(a[i])));return res;}



constexpr bool bget(ll m, int keta) {return (m >> keta) & 1;}

int bget(ll m, int keta, int sinsuu) {m /= (ll) pow(sinsuu, keta);return m % sinsuu;}

ll bit(int n) { return (1LL << (n)); }

ll bit(int n, int sinsuu) { return (ll) pow(sinsuu, n); }

int mask(int n) { return (1ll << n) - 1; }

#define bcou __builtin_popcountll

//最下位ビット

int lbit(int n) {return n & -n;}

//最上位ビット

int hbit(int n) {n |= (n >> 1);n |= (n >> 2);n |= (n >> 4);n |= (n >> 8);n |= (n >> 16);n |= (n >> 32);return n - (n >> 1);}

int hbitk(int n) {int k = 0;rer(i, 5) {int a = k + (1ll << i);int b = 1ll << a;if (b <= n)k += 1ll << i;}return k;}

//初期化は0を渡す

ll nextComb(ll &mask, int n, int r) {    if (!mask)return mask = (1LL << r) - 1;    ll x = mask & -mask; /*最下位の1*/    ll y = mask + x; /*連続した下の1を繰り上がらせる*/    ll res = ((mask & ~y) / x >> 1) | y;    if (bget(res, n))return mask = 0;    else return mask = res;}

//n桁以下でビットがr個立っているもののvectorを返す

vi bitCombList(int n, int r) {vi res;int m = 0;while (nextComb(m, n, r)) {res.push_back(m);}return res;}

char itoal(int i) {    return 'a' + i;}

char itoaL(int i) {    return 'A' + i;}

int altoi(char c) {    if ('A' <= c && c <= 'Z')return c - 'A';    return c - 'a';}

int ctoi(char c) { return c - '0'; }

char itoc(int i) { return i + '0'; }

int vtoi(vi &v) {    int res = 0;    if (sz(v) > 18) {        debugline("vtoi");        deb(sz(v));        ole();    }    rep(i, sz(v)) {        res *= 10;        res += v[i];    }    return res;}

vi itov(int i) {    vi res;    while (i) {        res.push_back(i % 10);        i /= 10;    }    rev(res);    return res;}

vi stov(string &a) {    int n = sz(a);    vi ret(n);    rep(i, n) {        ret[i] = a[i] - '0';    }    return ret;}

//基準を満たさないものは0になる

vi stov(string &a,char one) {    int n=sz(a);    vi ret (n);    rep(i,n)ret[i]=a[i]==one;    return ret;}

vector<vector<int>> ctoi(vector<vector<char>> s, char c) {    int n = sz(s), m = sz(s[0]);    vector<vector<int>> res(n, vector<int>(m));    rep(i, n)rep(j, m)res[i][j] = s[i][j] == c;    return res;}

#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );

//[i] := i番として圧縮されたものを返す

vi compress(vi &a) {    vi b;    int len = a.size();    for (int i = 0; i < len; ++i) {        b.push_back(a[i]);    }    sort(b);    unique(b);    for (int i = 0; i < len; ++i) {        a[i] = lower_bound(all(b), a[i]) - b.begin();    }    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vi &a, umap<int, int> &map) {    vi b;    int len = a.size();    for (int i = 0; i < len; ++i) {        b.push_back(a[i]);    }    sort(b);    unique(b);    for (int i = 0; i < len; ++i) {        int v = a[i];        a[i] = lower_bound(all(b), a[i]) - b.begin();        map[v] = a[i];    }    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vi &a, vi &r) {    vi b;    int len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    sort(b);    unique(b);    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vi &a, vi &r, vi &s) {    vi b;    int len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    fora(v, s)b.push_back(v);    sort(b);    unique(b);    for (int i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (int i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    for (int i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(V<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();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

vi compress(vector<vector<vi >> &a) {    vi b;    fora(vvv, a)fora(vv, vvv)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    int blen = sz(b);    vi ret(blen);    rep(i, blen) {        ret[i] = b[i];    }    return ret;}

void compress(int a[], int len) {    vi b;    for (int i = 0; i < len; ++i) {        b.push_back(a[i]);    }    sort(b);    unique(b);    for (int i = 0; i < len; ++i) {        a[i] = lower_bound(all(b), a[i]) - b.begin();    }}

//要素が見つからなかったときに困る

#define binarySearch(a, v) (binary_search(all(a),v))

#define lowerIndex(a, v) (lower_bound(all(a),v)-a.begin())

#define lowerBound(a, v) (*lower_bound(all(a),v))

#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())

#define upperBound(a, v) (*upper_bound(all(a),v))

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



//便利 数学 math

int mod(int a, int m) { return (a % m + m) % m; }

int pow(int a) { return a * a; };

ll fact(int v) { return v <= 1 ? 1 : v * fact(v - 1); }



ll comi(int n, int r) {    assert(n < 100);    static vvi(pas, 100, 100);    if (pas[0][0])return pas[n][r];    pas[0][0] = 1;    rep(i, 1, 100) {        pas[i][0] = 1;        rep(j, 1, i + 1)pas[i][j] = pas[i - 1][j - 1] + pas[i - 1][j];    }    return pas[n][r];}

double comd(int n, int r) {    static vd fac;    if (sz(fac) < n + 1) {        if (sz(fac) == 0)fac.push_back(1);        rep(i, sz(fac) - 1, n) {            fac.push_back(fac.back() * (i + 1));        }    }    if (n < r || n <= 0)return 0;    return fac[n] / fac[n - r] / fac[r];}

int gcd(int a, int b) {    while (b) a %= b, swap(a, b);    return abs(a);}

int gcd(vi b) {    ll res = b[0];    rep(i, 1, sz(b))res = gcd(b[i], res);    return res;}

ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }

ll lcm(vi a) {    int res = a[0];    rep(i, 1, sz(a))res = lcm(a[i], res);    return res;}

ll ceil(ll a, ll b) {    if (b == 0) {        debugline("ceil");        deb(a, b);        ole();        return -1;    } else if (a < 0) {        return 0;    } else {return (a + b - 1) / b;}}

int lower_remi__bx_a(int kei, int rem, int x) {    if (rem >= x) return 0;    return (x - rem + kei - 1) / kei;}

int lower_remv__bx_a(int kei, int rem, int x) {    if (rem >= x) return rem;    return (x - rem + kei - 1) / kei * kei + rem;}

int upper_remi__bx_a(int kei, int rem, int x) {    if (rem > x) return 0;    return (x - rem + kei) / kei;}

int upper_remv__bx_a(int kei, int rem, int x) {    if (rem > x) return rem;    return (x - rem + kei) / kei * kei + rem;}



//v * v >= aとなる最小のvを返す

ll sqrt(ll a) {if (a < 0) {debugline("sqrt");deb(a);ole();}ll res = (ll) std::sqrt(a);while (res * res < a)++res;return res;}

double log(double e, double x) { return log(x) / log(e); }

ll sig(ll t) { return (1 + t) * t / 2; }

ll sig(ll s, ll t) { return (s + t) * (t - s + 1) / 2; }





//幾何 Pをcomplexとして扱う

template<class T, class U> bool eq(T a, U b) { return fabs(a - b) < eps; }

dou atan2(pd a) { return atan2(a.se, a.fi); }

dou angle(pd f, pd t) { return atan2(t.se - f.se, t.fi - f.fi); }

dou distance(pd a, pd b) { return hypot(a.fi - b.fi, a.se - b.se); }

//bを中心とするabcのtheta aからcにかけて時計回り

dou angle(pd a, pd b, pd c) {    dou ax = a.fi - b.fi;    dou ay = a.se - b.se;    dou cx = c.fi - b.fi;    dou cy = c.se - b.se;    double ret = atan2(cy, cx) - atan2(ay, ax);    if (ret < 0) ret += 2 * PI;    return ret;}

dou dot(pd a, pd b) { return a.fi * b.fi + a.se + b.se; }

dou cro(pd a, pd b) { return a.fi * b.se - a.se + b.fi; }



//機能拡張

template<class T, class U> void operator+=(queue<T> &a, U v) { a.push(v); }

template<class T, class U> void operator+=(deque<T> &a, U v) { a.push_back(v); }

template<class T> 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> set<T> &operator+=(set<T> &a, U v) {    a.insert(v);    return a;}

template<class T, class U> set<T, greater<T>> &operator+=(set<T, greater<T>> &a, U v) {    a.insert(v);    return a;}

template<class T, class U> vector<T> &operator+=(vector<T> &a, U v) {    a.push_back(v);    return a;}

template<class T, class U> vector<T> operator+(const vector <T> &a, U v) {vector<T> ret = a;ret += v;return ret;}

template<class T, class U> vector<T> operator+(U v, const vector <T> &a) {vector<T> ret = a;ret.insert(ret.begin(), v);return ret;}

template<class T> vector<T> operator+(vector<T> a, vector <T> b) {vector<T> ret;ret = a;fora(v, b)ret += v;return ret;}

template<class T> vector<T> &operator+=(vector<T> &a, vector <T> &b) {fora(v, b)a += v;return a;}

template<class T> vector<T> &operator-=(vector<T> &a, vector <T> &b) {if (sz(a) != sz(b)) {debugline("vector<T> operator-=");deb(a);deb(b);exit(0);}rep(i, sz(a))a[i] -= b[i];return a;}

template<class T> vector<T> operator-(vector<T> &a, vector <T> &b) {if (sz(a) != sz(b)) {debugline("vector<T> operator-");deb(a);deb(b);ole();}vector<T> res(sz(a));rep(i, sz(a))res[i] = a[i] - b[i];return res;}

template<class T,class U> vector<T> operator*(vector<T> &a, U b) {vector<T> ret;fora(v,a)ret+=v*b;return ret;}

template<class T,class U> vector<T> operator/(vector<T> &a, U b) {vector<T> ret;fora(v,a)ret+=v/b;return ret;}

template<class T,class U> vector<T> operator*=(vector<T> &a, U b) {fora(v,a)v*=b;return a;}

template<class T,class U> vector<T> operator/=(vector<T> &a, U b) {fora(v,a)v/=b;return a;}

template<typename T> void erase(vector<T> &v, unsigned int i) {v.erase(v.begin()+ i);}

template<typename T> void erase(vector<T> &v, unsigned int s,unsigned int e) {v.erase(v.begin()+ s, v.begin()+ e);}

template<class T, class U> void erase(map<T, U> &m, int okl, int ngr) {m.erase(m.lower_bound(okl), m.lower_bound(ngr));}

template<class T> void erase(set<T> &m, int okl, int ngr) {m.erase(m.lower_bound(okl), m.lower_bound(ngr));}

template<typename T> void erasen(vector<T> &v, unsigned int s,unsigned int n) {v.erase(v.begin()+ s, v.begin()+ s + n);}

template<typename T, typename U> void insert(vector<T> &v, unsigned int i,U t) {v.insert(v.begin()+ i, t);}

template<typename T, typename U> void push_front(vector<T> &v, U t) {v.insert(v.begin(), t);}

template<typename T, typename U> void insert(vector<T> &v, unsigned int i,vector<T> list) {for (auto &&va:list)v.insert(v.begin()+ i++, va);}

template<typename T> void insert(set<T> &v, vector<T> list) {for (auto &&va :list)v.insert(va);}

vector<string> split(const string a, const char deli) {    string b = a + deli;    int l = 0, r = 0, n = b.size();    vector<string> res;    rep(i, n) {        if (b[i] == deli) {            r = i;            if (l < r)res.push_back(b.substr(l, r - l));            l = i + 1;        }    }    return res;}

vector<string> split(const string a, const string deli) {    vector<string> res;    int kn = sz(deli);    std::string::size_type Pos(a.find(deli));    int l = 0;    while (Pos != std::string::npos) {        if (Pos - l)res.push_back(a.substr(l, Pos - l));        l = Pos + kn;        Pos = a.find(deli, Pos + kn);    }    if (sz(a) - l)res.push_back(a.substr(l, sz(a) - l));    return res;}

void yn(bool a) {if (a)cout << "yes" << endl;    else cout << "no" << endl;}

void Yn(bool a) {if (a)cout << "Yes" << endl;    else cout << "No" << endl;}

void YN(bool a) {if (a)cout << "YES" << endl;    else cout << "NO" << endl;}

void fyn(bool a) {if (a)cout << "yes" << endl;    else cout << "no" << endl;    exit(0);}

void fYn(bool a) {if (a)cout << "Yes" << endl;    else cout << "No" << endl;    exit(0);}

void fYN(bool a) {if (a)cout << "YES" << endl;    else cout << "NO" << endl;    exit(0);}

void Possible(bool a) {if (a)cout << "Possible" << endl;    else cout << "Impossible" << endl;    exit(0);}

void POSSIBLE(bool a) {if (a)cout << "POSSIBLE" << endl;    else cout << "IMPOSSIBLE" << endl;    exit(0);}

//@起動時

struct initon {

    initon() {

        cin.tie(0);

        ios::sync_with_stdio(false);

        cout.setf(ios::fixed);

        cout.precision(16);

        srand((unsigned) clock() + (unsigned) time(NULL));

    };

} initonv;



//@formatter:on

//gra mint pr

int n, m, k, d, H, W, x, y, z, q;

int cou;

vi t, a, b, c;

//vvi (s, 0, 0);

vvc (ba, 0, 0);

vp p;

str s;





void solve() {

    in(n);

    vi x, y;

    vc d;

    na3(x, y, d, n);

    auto cal = [&](dou v) {

        dou mix = linf;

        dou miy = linf;

        dou max = -linf;

        dou may = -linf;

        rep(i, n) {

            dou nx = x[i], ny = y[i];

            if (d[i] == 'R')nx += v;

            elf (d[i] == 'L')nx -= v;

            elf (d[i] == 'U')ny += v;

            elf (d[i] == 'D')ny -= v;

            chmi(mix, nx);

            chma(max, nx);

            chmi(miy, ny);

            chma(may, ny);

        }

        return (max - mix) * (may - miy);

    };

    dou i = golddd_s(0.0, 1e18, cal, 5000);

    dou best = cal(i);

    cout << best << endl;

}

int my(int n, vi &a) {

    return 0;

}

int sister(int n, vi &a) {

    int ret = 0;

    return ret;

}



signed main() {

    solve();



#define arg n,a

#ifdef _DEBUG

    bool bad = 0;

    for (int i = 0, ok = 1; i < k5 && ok; ++i) {

        int n = rand(1, 8);

        vi a = ranv(n, 1, 10);

        int myres = my(arg);

        int res = sister(arg);

        ok = myres == res;

        if (!ok) {

            out(arg);

            cerr << "正解 : " << res << endl;

            cerr << "自分  : " << myres << endl;

            bad = 1;

            break;

        }

    }

    if (!bad) {

//        cout << "完璧 : solveを書き直そう" << endl;

//        cout << "     : そして、solve()を呼び出すのだ" << endl;

//        cout << "     : cin>>n; na(a,n);も忘れるな" << endl;

    }

#endif

    return 0;

};




// Fastest known solution for this problem: