//#pragma GCC optimize("Ofast")
#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace std::chrono;
#define int long long
#define ll long long
auto start_time = system_clock::now();

//@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(ll k) {        auto it = s.begin();        for (ll i = 0; i < k; i++)it++;        return it;    }    auto order_of_key(ll v) {        auto it = s.begin();        ll i=0;        for (;it != s.end() && *it <v ; i++)it++;        return i;    }};
#define pbds(T) my_pbds_tree<T>
#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 {        ll v=((x.first) << 32) | x.second;        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();        return splitmix64(v + FIXED_RANDOM);    }};
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;    }    ll operator[](ll i) {        assert(i < 4);        return i == 0 ? a : i == 1 ? b : i == 2 ? c : d;    }};
typedef T2<ll, ll, ll> T;
typedef F2<ll, ll, ll, ll> F;
T mt(ll a, ll b, ll c) {return T(a, b, c);}

//@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<ll, ll>;
#define fi first
#define se second
#define beg begin
#define rbeg rbegin
#define con continue
#define bre break
#define brk break
#define is ==
#define el else
#define elf else if
#define wh while
#define upd update

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

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

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

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

template<typename T> vector<T> make_v(size_t a) { return vector<T>(a); }
template<typename T, typename... Ts> auto make_v(size_t a, Ts... ts) {return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));}
#define vni(name, ...) auto name = make_v<ll>(__VA_ARGS__)
#define vnb(name, ...) auto name = make_v<bool>(__VA_ARGS__)
#define vns(name, ...) auto name = make_v<string>(__VA_ARGS__)
#define vnd(name, ...) auto name = make_v<double>(__VA_ARGS__)
#define vnc(name, ...) auto name = make_v<char>(__VA_ARGS__)
#define vnp(name, ...) auto name = make_v<P>(__VA_ARGS__)

#define PQ priority_queue<ll, vector<ll>, greater<ll> >
#define tos to_string
using mapi = map<ll, ll>;
using mapp = map<P, ll>;
using mapd = map<dou, ll>;
using mapc = map<char, ll>;
using maps = map<str, ll>;
using seti = set<ll>;
using setd = set<dou>;
using setc = set<char>;
using sets = set<str>;
using qui = queue<ll>;
#define bset bitset
#define uset unordered_set
#define useti unordered_set<ll,ll,xorshift>
#define mset multiset
#define mseti multiset<ll>
#define umap unordered_map
#define umapi unordered_map<ll,ll,xorshift>
#define umapp unordered_map<P,ll,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;}    ll size() {return q.size();}};
template<class T> struct pqg {    priority_queue<T> q;/*大きい順*/    T su = 0;    void clear() {q = priority_queue<T>();su = 0;}    void operator+=(T v) {su += v;q.push(v);}    T sum() {return su;}    T top() {return q.top();}    void pop() {su -= q.top();q.pop();}    T poll() {T ret = q.top();su -= ret;q.pop();return ret;}    ll size() {return q.size();}};
#define pqi pq<ll>
#define pqgi pqg<ll>
//マクロ 繰り返し
#define o_rep(o1, o2, o3, o4, name, ...) name
# define rep1(n) for(ll rep1i = 0,rep1lim=n; rep1i < rep1lim ; ++rep1i)
# define rep2(i, n) for(ll i = 0,rep2lim=n; i < rep2lim ; ++i)
#define rep3(i, m, n) for(ll i = m,rep3lim=n; i < rep3lim ; ++i)
#define rep4(i, m, n, ad) for(ll i = m,rep4lim=n; i < rep4lim ; i+= ad)
#define rep(...) o_rep(__VA_ARGS__,rep4,rep3,rep2,rep1)(__VA_ARGS__)

#define rer2(i, n) for(ll i = n; i >= 0 ; i--)
#define rer3(i, m, n) for(ll i = m,rer3lim=n; i >= rer3lim ; i--)
#define rer4(i, m, n, dec) for(ll i = m,rer4lim=n; i >= rer4lim ; i-=dec)
#define rer(...) o_rep(__VA_ARGS__,rer4,rer3,rer2,)(__VA_ARGS__)

#define reps2(i, j, n) for(ll i = 0,reps2lim=n; i < reps2lim ;++i)for(ll j = 0; j < reps2lim ; ++j)
#define reps3(i, j, k, n) for(ll i = 0,reps3lim=n; i < reps3lim ; ++i)for(ll j = 0; j < reps3lim ; ++j)for(ll k = 0; k < reps3lim ; ++k)
#define reps4(i, j, k, l, n) for(ll i = 0,reps4lim=n; i < reps4lim ; ++i)for(ll j = 0; j < reps4lim ; ++j)for(ll k = 0; k < reps4lim ; ++k)for(ll 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(ll i = 0; i < a ; ++i)for(ll j = 0; j < b ; ++j)for(ll k = 0; k < c ; ++k)

#define fora(a, b) for(auto&& a : b)
//#define fori(i, s, len) for (int i = s, prev = (s == 0) ? len - 1 : s - 1, next = (s == len - 1) ? 0 : s + 1, cou = 0; cou < len; cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
//インデックスを前後含めて走査
#define fori(i, s, len) for (int i = s, prev = (s == 0) ? len - 1 : s - 1, next = (s == len - 1) ? 0 : s + 1, cou = 0; cou < len; cou++, prev = i, i = next, next = (next == len - 1) ? 0 : next + 1)
//vectorの中身を先頭から見る
#define foriv(i, v, d) int i=0;for (auto prev = d[sz(d)-1],next=d[1],v = d[0]; i < sz(d); i++, prev = v, v = next, next = (i>=sz(d)-1?d[0]:d[i+1]))

#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 ll inf = (ll) 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();

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

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

//埋め込み 存在を意識せずに機能を増やされているもの
//@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 (ll i = 0; i < vec.size(); ++i)os << vec[i] << (i + 1 == vec.size() ? "" : " ");    return os;}template<typename T> ostream &operator<<(ostream &os, vector<vector<T>> &vec) {    for (ll i = 0; i < vec.size(); ++i) {        for (ll j = 0; j < vec[i].size(); ++j) { os << vec[i][j] << " "; }        os << endl;    }    return os;}template<typename T, typename U> ostream &operator<<(ostream &os, map<T, U> &m) {    for (auto &&v:m) os << v;    return os;}template<class T> ostream &operator<<(ostream &os, set<T>s) {fora(v,s){os<<v<<" ";}return os;}template<class T> ostream &operator<<(ostream &os, deque<T> a) {fora(v,a)os<<v<<" ";return os;}
    template<typename W, typename H> void resize(vector<W> &vec, const H head) { vec.resize(head); }template<typename W, typename H, typename ... T> void resize(vector<W> &vec, const H &head, const T ... tail) {vec.resize(head);for (auto &v: vec)resize(v, tail...);}
template<typename T, typename F> bool all_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool all_of2(vector<T> &v, F f) {    rep(i, sz(v)) { if (!all_of2(v[i], f))return false; }    return true;}
template<typename T, typename F> bool any_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool any_of2(vector<T> &v, F f) {    rep(i, sz(v)) { if (any_of2(v[i], f))return true; }    return false;}
template<typename T, typename F> bool none_of2(T &v, F f) { return f(v); }
template<typename T, typename F> bool none_of2(vector<T> &v, F f) {    rep(i, sz(v)) { if (none_of2(v[i], f))return false; }    return true;}
template<typename T, typename F> bool find_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll find_if2(vector<T> &v, F f) {    rep(i, sz(v)) { if (find_if2(v[i], f))return i; }    return sz(v);}
template<typename T, typename F> bool rfind_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll rfind_if2(vector<T> &v, F f) {    rer(i, sz(v) - 1) { if (rfind_if2(v[i], f))return i; }    return -1;}
template<class T> bool contains(string &s, const T &v) { return s.find(v) != string::npos; }
template<typename T> bool contains(vector<T> &v, const T &val) { return std::find(v.begin(), v.end(), val) != v.end(); }
template<typename T, typename F> bool contains_if2(vector<T> &v, F f) { return find_if(v.begin(), v.end(), f) != v.end(); }
template<typename T, typename F> ll count_if2(T &v, F f) { return f(v); }
template<typename T, typename F> ll count_if2(vector<T> &vec, F f) {    ll ret = 0;    fora(v, vec)ret += count_if2(v, f);    return ret;}
template<typename T, typename F> void for_each2(T &v, F f) { f(v); }
template<typename T, typename F> void for_each2(vector<T> &vec, F f) { fora(v, vec)for_each2(v, f); }
template<typename W> ll count_od(vector<W> &a) {return count_if2(a,[](ll v){return v&1 ;});}
template<typename W> ll count_ev(vector<W> &a) {return count_if2(a,[](ll v){return !(v&1) ;});}
#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;ll kn = sz(key);std::string::size_type Pos(a.find(key));ll l = 0;while (Pos != std::string::npos) {t << a.substr(l, Pos - l);t << v;l = Pos + kn;Pos = a.find(key, Pos + kn);}t << a.substr(l, sz(a) - l);a = t.str();}
template<class T> bool includes(vector<T> &a, vector<T> &b) {vi c = a;vi d = b;sort(all(c));sort(all(d));return includes(all(c), all(d));}
template<class T> bool is_permutation(vector<T> &a, vector<T> &b) { return is_permutation(all(a), all(b)); }
template<class T> bool next_permutation(vector<T> &a) { return next_permutation(all(a)); }
void iota(vector<ll> &ve, ll s, ll n) {ve.resize(n);iota(all(ve), s);}
vi iota(ll s, ll len) {vi ve(len);iota(all(ve), s);return ve;}
template<class A, class B> auto vtop(vector<A> &a, vector<B> &b) {    assert(sz(a) == sz(b));    /*stringを0で初期化できない  */  vector<pair<A, B>> res;    rep(i, sz(a))res.eb(a[i], b[i]);return res;}
template<class A, class B> void ptov(vector<pair<A, B>> &p, vector<A> &a, vector<B> &b) {    a.resize(sz(p)), b.resize(sz(p));    rep(i, sz(p))a[i] = p[i].fi, b[i] = p[i].se;}
template<class A, class B, class C> auto vtot(vector<A> &a, vector<B> &b, vector<C> &c) {    assert(sz(a) == sz(b) && sz(b) == sz(c));    vector<T2<A, B, C>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i]);    return res;}
template<class A, class B, class C, class D> auto vtof(vector<A> &a, vector<B> &b, vector<C> &c, vector<D> &d) {    assert(sz(a) == sz(b) && sz(b) == sz(c) && sz(c) == sz(d));    vector<F2<A, B, C, D>> res;    rep(i, sz(a))res.eb(a[i], b[i], c[i], d[i]);    return res;}
enum pcomparator { fisi, fisd, fdsi, fdsd, sifi, sifd, sdfi, sdfd };
enum tcomparator {    fisiti, fisitd, fisdti, fisdtd, fdsiti, fdsitd, fdsdti, fdsdtd,    fitisi, fitisd, fitdsi, fitdsd, fdtisi, fdtisd, fdtdsi, fdtdsd,    sifiti, sifitd, sifdti, sifdtd, sdfiti, sdfitd, sdfdti, sdfdtd,    sitifi, sitifd, sitdfi, sitdfd, sdtifi, sdtifd, sdtdfi, sdfdfd,    tifisi, tifisd, tifdsi, tifdsd, tdfisi, tdfisd, tdfdsi, tdfdsd,    tisifi, tisifd, tisdfi, tisdfd, tdsifi, tdsifd, tdsdfi, tdsdfd};
template<class A, class B> void sort(vector<pair<A, B>> &a, pcomparator type) {    typedef pair<A, B> U;    if (type == fisi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se < r.se; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi < r.fi : l.se > r.se; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se < r.se; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.fi != r.fi ? l.fi > r.fi : l.se > r.se; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi < r.fi; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se < r.se : l.fi > r.fi; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi < r.fi; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.se != r.se ? l.se > r.se : l.fi > r.fi; });};template<class U> void sort(vector<U> &a, pcomparator type) {    if (type == fisi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f > r.f; });};template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D> > &a, pcomparator type) {    typedef F2<A, B, C, D> U;    if (type == fisi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == fisd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == fdsi) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == fdsd) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == sifi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == sifd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == sdfi) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == sdfd) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a > r.a; });};template<class U> void sort(vector<U> &a, tcomparator type) {    if (type == 0) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });    else if (type == 1) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });    else if (type == 2) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });    else if (type == 3) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });    else if (type == 4) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t < r.t; });    else if (type == 5) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s < r.s : l.t > r.t; });    else if (type == 6) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t < r.t; });    else if (type == 7) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.s != r.s ? l.s > r.s : l.t > r.t; });    else if (type == 8) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });    else if (type == 9) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });    else if (type == 10) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });    else if (type == 11) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f < r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });    else if (type == 12) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s < r.s; });    else if (type == 13) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t < r.t : l.s > r.s; });    else if (type == 14) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s < r.s; });    else if (type == 15) sort(all(a), [&](U l, U r) { return l.f != r.f ? l.f > r.f : l.t != r.t ? l.t > r.t : l.s > r.s; });    else if (type == 16) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });    else if (type == 17) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });    else if (type == 18) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });    else if (type == 19) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });    else if (type == 20) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t < r.t; });    else if (type == 21) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f < r.f : l.t > r.t; });    else if (type == 22) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t < r.t; });    else if (type == 23) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.f != r.f ? l.f > r.f : l.t > r.t; });    else if (type == 24) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });    else if (type == 25) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });    else if (type == 26) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });    else if (type == 27) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s < r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });    else if (type == 28) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f < r.f; });    else if (type == 29) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t < r.t : l.f > r.f; });    else if (type == 30) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f < r.f; });    else if (type == 31) sort(all(a), [&](U l, U r) { return l.s != r.s ? l.s > r.s : l.t != r.t ? l.t > r.t : l.f > r.f; });    else if (type == 32) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == 33) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == 34) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == 35) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == 36) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s < r.s; });    else if (type == 37) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f < r.f : l.s > r.s; });    else if (type == 38) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s < r.s; });    else if (type == 39) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.f != r.f ? l.f > r.f : l.s > r.s; });    else if (type == 40) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == 41) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == 42) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == 43) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t < r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });    else if (type == 44) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f < r.f; });    else if (type == 45) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s < r.s : l.f > r.f; });    else if (type == 46) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f < r.f; });    else if (type == 47) sort(all(a), [&](U l, U r) { return l.t != r.t ? l.t > r.t : l.s != r.s ? l.s > r.s : l.f > r.f; });}template<class A, class B, class C, class D> void sort(vector<F2<A, B, C, D>> &a, tcomparator type) {    typedef F2<A, B, C, D> U;    if (type == 0) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });    else if (type == 1) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });    else if (type == 2) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });    else if (type == 3) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });    else if (type == 4) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c < r.c; });    else if (type == 5) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b < r.b : l.c > r.c; });    else if (type == 6) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c < r.c; });    else if (type == 7) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.b != r.b ? l.b > r.b : l.c > r.c; });    else if (type == 8) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });    else if (type == 9) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });    else if (type == 10) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });    else if (type == 11) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a < r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });    else if (type == 12) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b < r.b; });    else if (type == 13) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c < r.c : l.b > r.b; });    else if (type == 14) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b < r.b; });    else if (type == 15) sort(all(a), [&](U l, U r) { return l.a != r.a ? l.a > r.a : l.c != r.c ? l.c > r.c : l.b > r.b; });    else if (type == 16) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });    else if (type == 17) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });    else if (type == 18) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });    else if (type == 19) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });    else if (type == 20) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c < r.c; });    else if (type == 21) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a < r.a : l.c > r.c; });    else if (type == 22) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c < r.c; });    else if (type == 23) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.a != r.a ? l.a > r.a : l.c > r.c; });    else if (type == 24) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });    else if (type == 25) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });    else if (type == 26) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });    else if (type == 27) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b < r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });    else if (type == 28) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a < r.a; });    else if (type == 29) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c < r.c : l.a > r.a; });    else if (type == 30) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a < r.a; });    else if (type == 31) sort(all(a), [&](U l, U r) { return l.b != r.b ? l.b > r.b : l.c != r.c ? l.c > r.c : l.a > r.a; });    else if (type == 32) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == 33) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == 34) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == 35) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == 36) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b < r.b; });    else if (type == 37) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a < r.a : l.b > r.b; });    else if (type == 38) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b < r.b; });    else if (type == 39) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.a != r.a ? l.a > r.a : l.b > r.b; });    else if (type == 40) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == 41) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == 42) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == 43) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c < r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });    else if (type == 44) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a < r.a; });    else if (type == 45) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b < r.b : l.a > r.a; });    else if (type == 46) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a < r.a; });    else if (type == 47) sort(all(a), [&](U l, U r) { return l.c != r.c ? l.c > r.c : l.b != r.b ? l.b > r.b : l.a > r.a; });}

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

template<typename A, size_t N, typename T> void fill(A (&a)[N], const T &v) { rep(i, N)a[i] = v; }template<typename A, size_t N, size_t O, typename T> void fill(A (&a)[N][O], const T &v) { rep(i, N)rep(j, O)a[i][j] = v; }template<typename A, size_t N, size_t O, size_t P, typename T> void fill(A (&a)[N][O][P], const T &v) { rep(i, N)rep(j, O)rep(k, P)a[i][j][k] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, typename T> void fill(A (&a)[N][O][P][Q], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)a[i][j][k][l] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, typename T> void fill(A (&a)[N][O][P][Q][R], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)a[i][j][k][l][m] = v; }template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S, typename T> void fill(A (&a)[N][O][P][Q][R][S], const T &v) { rep(i, N)rep(j, O)rep(k, P)rep(l, Q)rep(m, R)rep(n, S)a[i][j][k][l][m][n] = v; }
template<typename W, typename T>void fill(W &xx, const T vall) {    xx = vall;}template<typename W, typename T>void fill(vector<W> &vecc, const T vall) {    for (auto &&vx     : vecc)fill(vx, vall);}
template<typename W,typename T>void fill(vector<W> &xx,const T v,ll len) {rep(i, len)xx[i]=v;}
template<typename W,typename T>void fill(vector<vector<W>> &xx,const T v,ll lh,ll lw) {rep(i, lh)rep(j,lw)xx[i][j]=v;}
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
ll in() {ll ret;cin >> ret;return ret;}
string sin() {string ret;cin >> ret;return ret;}
template<class T>  void in(T &head) { cin >> head; }template<class T, class... U>  void in(T &head, U &... tail) {cin >> head;in(tail...);}

#define o_din(o1, o2, o3, o4, o5, o6, name, ...) name
#define din1(a) ll a;cin>>a
#define din2(a, b) ll a,b;cin>>a>> b
#define din3(a, b, c) ll a,b,c;cin>>a>>b>>c
#define din4(a, b, c, d) ll a,b,c,d;cin>>a>>b>>c>>d
#define din5(a, b, c, d, e) ll a,b,c,d,e;cin>>a>>b>>c>>d>>e
#define din6(a, b, c, d, e, f) ll a,b,c,d,e,f;cin>>a>>b>>c>>d>>e>>f
#define din(...) o_din(__VA_ARGS__,din6,din5,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__)

template<class T> void out2(T &&head) { cout << head; }
template<class T, class... U> void out2(T &&head, U &&... tail) {    cout << head << " ";    out2(tail...);}
template<class T, class... U> void out(T &&head, U &&... tail) {    cout << head << " ";    out2(tail...);    cout << "" << endl;}
template<class T> void out(T &&head) {    cout << head  << endl;}
void out() {    cout << ""  << endl;}
#ifdef _DEBUG
template<class T> string out_m(vector<T> &a, ll W = inf) {stringstream ss;    if (W == inf)W = min(sz(a), 12ll);   if(sz(a)==0)return ss.str();   rep(i, W) { ss << a[i] << " "; }    ss << "" << endl;    return ss.str();}
template<class T> string out_m(vector<vector<T> > &a, ll H = inf, ll W = inf, int key = -1) {
    H = min({H, sz(a), 12ll});
    W = min({W, sz(a[0]), 12ll});
    stringstream ss;
    ss << endl;
    if (key == -1)ss << " *|"; else ss << " " << key << "|";
    rep(w, W)ss << std::right << std::setw(4) << w;
    ss << "" << endl;
    rep(w, W * 4 + 3)ss << "_";
    ss << "" << endl;
    rep(h, H) {
        ss << std::right << std::setw(2) << h << "|";
        rep(w, min(sz(a[h]),12ll)) { if (a[h][w] == linf) ss << "  e" << " "; else ss << std::right << std::setw(4) << a[h][w]; }
        ss << "" << endl;
    }
    ss << endl;
    return ss.str();
}
/*@formatter:off*/
template<class T> string out_m(vector<vector<vector<T> > > &a, ll H = inf, ll W = inf, ll U = inf) {stringstream ss;    if (H == inf)H = 5;    H = min(H, sz(a));    rep(i, H) {        ss << endl;        ss << out_m(a[i], W, U, i);    }    ss << endl;    return ss.str();}
string out_m(int a) {stringstream ss;ss << a << endl;return ss.str();}
template<class T> string out_m(T &a) {stringstream ss;ss << a << endl;return ss.str();}
template<class T> void outv(vector<T> &a, ll W=inf) {cout << out_m(a,W) << endl;}
template<class T> void outv(vector<vector<T> > &a, ll H = linf, ll W = linf,int key=-1) {    cout << out_m(a,H,W,key) << endl;}
template<class T> void outv(vector<vector<vector<T> > > &a, ll H = linf, ll W = linf,ll U = linf) {cout << out_m(a,H,W,U)<< endl;}
#else
template<class T> void outv(vector<T> &a, ll W = inf) {
    rep(i, min(W, sz(a))) { cout << a[i] << " "; }
    cout << "" << endl;
}
template<class T> void outv(vector<vector<T> > &a, ll H = linf, ll W = linf, int key = -1) { rep(i, min(H, sz(a))) { outv(a[i], W); }}
template<class T> void outv(vector<vector<vector<T> > > &a, ll H = linf, ll W = linf, ll U = linf) { ; }
#endif
template<class T> void outl(vector<T> &a, int n = inf) { rep(i, min(n, sz(a)))cout << a[i] << endl; }
template<class T> void na(vector<T> &a, ll n) {
    a.resize(n);
    rep(i, n)cin >> a[i];
}
#define dna(a, n) vi a(n); rep(dnai,n) cin >> a[dnai];
template<class T> void nao(vector<T> &a, ll n) {
    a.resize(n + 1);
    a[0] = 0;
    rep(i, n)cin >> a[i + 1];
}
template<class T> void naod(vector<T> &a, ll n) {
    a.resize(n + 1);
    a[0] = 0;
    rep(i, n)cin >> a[i + 1], a[i + 1]--;
}
template<class T> void nad(vector<T> &a, ll n) {
    a.resize(n);
    rep(i, n)cin >> a[i], a[i]--;
}
template<class T, class U> void na2(vector<T> &a, vector<U> &b, ll n) {
    a.resize(n);
    b.resize(n);
    rep(i, n)cin >> a[i] >> b[i];
}
#define dna2(a, b, n) vi a(n),b(n);rep(dna2i, n)cin >> a[dna2i] >> b[dna2i];
template<class T, class U> void nao2(vector<T> &a, vector<U> &b, ll n) {
    a.resize(n + 1);
    b.resize(n + 1);
    a[0] = b[0] = 0;
    rep(i, n)cin >> a[i + 1] >> b[i + 1];
}
#define dna2d(a, b, n) vi a(n),b(n);rep(dna2di, n){cin >> a[dna2di] >> b[dna2di];a[dna2di]--,b[dna2di]--;}
template<class T, class U> void na2d(vector<T> &a, vector<U> &b, ll n) {
    a.resize(n);
    b.resize(n);
    rep(i, n)cin >> a[i] >> b[i], a[i]--, b[i]--;
}
template<class T, class U, class W> void na3(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
    a.resize(n);
    b.resize(n);
    c.resize(n);
    rep(i, n)cin >> a[i] >> b[i] >> c[i];
}
#define dna3(a, b, c, n) vi a(n),b(n),c(n);   rep(dna3i, n)cin >> a[dna3i] >> b[dna3i] >> c[dna3i];
template<class T, class U, class W> void na3d(vector<T> &a, vector<U> &b, vector<W> &c, ll n) {
    a.resize(n);
    b.resize(n);
    c.resize(n);
    rep(i, n)cin >> a[i] >> b[i] >> c[i], a[i]--, b[i]--, c[i]--;
}
#define dna3d(a, b, c, n) vi a(n),b(n),c(n);  rep(dna3di, n){cin >> a[dna3di] >> b[dna3di] >> c[dna3di];a[dna3di]--,b[dna3di]--,c[dna3di]--;}
template<class T, class U, class W, class X> void na4(vector<T> &a, vector<U> &b, vector<W> &c, vector<X> &d, ll n) {
    a.resize(n);
    b.resize(n);
    c.resize(n);
    d.resize(n);
    rep(i, n)cin >> a[i] >> b[i] >> c[i] >> d[i];
}
#define dna4(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   rep(dna4i, n)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i];
#define dna4d(a, b, c, d, n) vi a(n),b(n),c(n),d(n);   rep(dna4i, n)cin >> a[dna4i] >> b[dna4i] >> c[dna4i]>>d[dna4i],--a[dna4i] ,-- b[dna4i],-- c[dna4i],--d[dna4i];
#define nt(a, h, w) resize(a,h,w);rep(nthi,h)rep(ntwi,w) cin >> a[nthi][ntwi];
#define ntd(a, h, w) resize(a,h,w);rep(ntdhi,h)rep(ntdwi,w) cin >> a[ntdhi][ntdwi], a[ntdhi][ntdwi]--;
#define ntp(a, h, w) resize(a,h+2,w+2);fill(a,'#');rep(ntphi,1,h+1)rep(ntpwi,1,w+1) cin >> a[ntphi][ntpwi];
//デバッグ
#define sp << " " <<

#define debugName(VariableName) # VariableName

#define deb1(x)  debugName(x)<<" = "<<out_m(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:off
//よく使うクラス、構造体
struct unionfind {
    vector<ll> par;
    vector<ll> siz;
    vector<ll> es;
    ll n, trees;//連結グループの数(親の種類)
    unionfind(ll n) : n(n), trees(n) {        par.resize(n);        siz.resize(n);        es.resize(n);        for (ll i = 0; i < n; i++) {            par[i] = i;            siz[i] = 1;        }    }
    ll root(ll x) { if (par[x] == x) { return x; } else { return par[x] = root(par[x]); }}
    bool unite(ll x, ll y) {
        x = root(x);
        y = root(y);
        es[x]++;
        if (x == y) return false;
        if (siz[x] > siz[y]) swap(x, y);
        trees--;
        par[x] = y;
        siz[y] += siz[x];
        es[y] += es[x];
        return true;
    }
    bool same(ll x, ll y) { return root(x) == root(y); }
    ll size(ll x) { return siz[root(x)]; }
    ll esize(ll x) { return es[root(x)]; }
    vi sizes(){        vi cou(n);        vi ret;        ret.reserve(n);        rep(i, n){            cou[root (i)]++;        }        rep(i, n){            if(cou[i])ret.push_back(cou[i]);        }        return ret;    }
    //つながりを無向グラフと見なし、xが閉路に含まれるか判定
    bool close(ll x) { return esize(x) >= size(x); }
    vec<vi> sets() {        vi ind(n, -1);        ll i = 0;        vvi(res, trees);        rep(j, n) {            ll r = root(j);            if (ind[r] == -1)ind[r] = i++;            res[ind[r]].push_back(j);        }        rep(i, trees) {            ll r = root(res[i][0]);            if (res[i][0] == r)continue;            rep(j, 1, sz(res[i])) {                if (res[i][j] == r) {                    swap(res[i][0], res[i][j]);                    break;                }            }        }        return res;    }
};//@formatter:off


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

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


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

//便利関数

//テスト用
char ranc() { return (char) ('a' + rand() % 26); }
ll rand(ll min, ll max) {    assert(min <= max);    if (min >= 0 && max >= 0) { return rand() % (max + 1 - min) + min; } else if (max < 0) { return -rand(-max, -min); } else { if (rand() % 2) { return rand(0, max); } else { return -rand(0, -min); }}}
vi ranv(ll n, ll min, ll max) {    vi v(n);    rep(i, n)v[i] = rand(min, max);    return v;}
str ransu(ll n) {    str s;    rep(i, n)s += (char) rand('A', 'Z');    return s;}
str ransl(ll n) {    str s;    rep(i, n)s += (char) rand('a', 'z');    return s;}
//単調増加
vi ranvinc(ll n, ll min, ll max) {    vi v(n);    bool bad = 1;    while (bad) {        bad = 0;        v.resize(n);        rep(i, n) {            if (i && min > max - v[i - 1]) {                bad = 1;                break;            }            if (i)v[i] = v[i - 1] + rand(min, max - v[i - 1]); else v[i] = rand(min, max);        }    }    return v;}
//便利 汎用

void ranvlr(ll n, ll min, ll max, vi &l, vi &r) {    l.resize(n);    r.resize(n);    rep(i, n) {        l[i] = rand(min, max);        r[i] = l[i] + rand(0, max - l[i]);    }}
vp run_length(vi &a) {    vp ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}    return ret;}
vector<pair<char, ll>> run_length(string &a) {    vector<pair<char, ll>> ret;    ret.eb(a[0], 1);    rep(i, 1, sz(a)) { if (ret.back().fi == a[i]) { ret.back().se++; } else { ret.eb(a[i], 1); }}    return ret;}
template<class F> ll mgr(ll ok, ll ng, F f) {    if (ok < ng)        while (ng - ok > 1) {            ll mid = (ok + ng) / 2;            if (f(mid))ok = mid; else ng = mid;        }    else        while (ok - ng > 1) {            ll 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 if (a < b)return b;    else return a;}
//strを整数として比較
string smin(str &a, str b) {    if (sz(a) > sz(b)) { return b; }    else if (sz(a) < sz(b)) { return a; }    else if (a > b)return b;    else return a;}
template<typename W, typename T> ll find(vector<W> &a, const T key) {    rep(i, sz(a))if (a[i] == key)return i;    return -1;}
template<typename W, typename T> P find(vector<vector<W >> &a, const T key) {    rep(i, sz(a))rep(j, sz(a[0]))if (a[i][j] == key)return mp(i, j);    return mp(-1, -1);}
template<typename W, typename U> T find(vector<vector<vector<W >>> &a, const U key) {    rep(i, sz(a))rep(j, sz(a[0]))rep(k, sz(a[0][0]))if (a[i][j][k] == key)return mt(i, j, k);    return mt(-1, -1, -1);}


template<typename W, typename T> ll count2(W &a, const T k) { return a == k; }
template<typename W, typename T> ll count2(vector<W> &a, const T k) {    ll ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
template<typename W, typename T> ll count(vector<W> &a, const T k) {    ll ret = 0;    fora(v, a)ret += count2(v, k);    return ret;}
ll count(str &a, str k) {    ll ret = 0, len = k.length();    auto pos = a.find(k);    while (pos != string::npos)pos = a.find(k, pos + len), ++ret;    return ret;}
vi count(str &a) {    vi cou(26);    char c = 'a';    if ('A' <= a[0] && a[0] <= 'Z')c = 'A';    rep(i, sz(a))++cou[a[i] - c];    return cou;}
#define couif count_if
//algorythm


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

ll get(ll a, ll keta) { return (a / (ll) pow(10, keta)) % 10; }
ll 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; }}}}
ll dsum(ll v,ll sin=10) {    ll ret = 0;    for (; v; v /= sin)ret += v % sin;    return ret;}

struct sint {
    ll v;
    sint(ll v) : v(v) {}
    operator ll() { return v; }
    //下からi番目
    ll operator[](ll i) { return (v / p10[i]) % 10; }
    ll back(ll i) { return operator[](i); }
    //上からi番目
    ll top(ll i) {
        ll len = keta(v);
        return operator[](len - 1 - i);
    }
    //先頭からi番目にセット
    ll settop(ll i, ll k) {
        ll len = keta(v);
        return set(len - 1 - i, k);
    }
    ll set(ll i, ll k) {
        if (i < 0)return settop(abs(i) - 1, k);
        return v += p10[i] * (k - (v / p10[i]) % 10);
    }
    ll add(ll i, ll k = 1) { return v += p10[i] * k; }
    ll addtop(ll i, ll k = 1) { return v += p10[keta(v) - i - 1] * k; }
    ll dec(ll i, ll k = 1) { return v -= p10[i] * k; }
    ll dectop(ll i, ll k = 1) { return v -= p10[keta(v) - i - 1] * k; }
#define op(t, o)template<class T> t operator o(T r){return v o r;}
    op(ll, +=);    op(ll, -=);    op(ll, *=);    op(ll, /=);    op(ll, %=);    op(ll, +);    op(ll, -);    op(ll, *);    op(ll, /);    op(ll, %);    op(bool, ==);    op(bool, !=);    op(bool, <);    op(bool, <=);    op(bool, >);    op(bool, >=);
#undef op
    template<class T>  ll operator<<=(T r) { return v *= p10[r]; }
    template<class T>  ll operator<<(T r) { return v * p10[r]; }
    template<class T>  ll operator>>=(T r) { return v /= p10[r]; }
    template<class T>  ll operator>>(T r) { return v / p10[r]; }
};
ll mask10(ll 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>  bool chma(T &a, const U &b) {    if (a < b) {        a = b;        return true;    }    return false;}
template<class U>  bool chma(const U &b) { return chma(ma, b); }
template<class T, class U>  bool chmi(T &a, const U &b) {    if (b < a) {        a = b;        return true;    }    return false;}
template<class U>  bool chmi(const U &b) { return chmi(mi, b); }
template<class T>  T min(T a, signed b) { return a < b ? a : b; }
template<class T>  T max(T a, signed b) { return a < b ? b : a; }
template<class T>  T min(T a, T b, T c) { return a >= b ? b >= c ? c : b : a >= c ? c : a; }
template<class T>  T max(T a, T b, T c) { return a <= b ? b <= c ? c : b : a <= c ? c : a; }
template<class T>  T min(vector<T>& a) { return *min_element(all(a)); }
template<class T>  T mini(vector<T>& a) { return min_element(all(a)) - a.begin(); }
template<class T>  T min(vector<T>& a, ll n) { return *min_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T min(vector<T>& a, ll s, ll n) { return *min_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a) { return *max_element(all(a)); }
template<class T,class U>  T max(vector<T>& a,vector<U>& b) { return max(*max_element(all(a)),*max_element(all(b))); }
template<class T>  T maxi(vector<T>& a) { return max_element(all(a)) - a.begin(); }
template<class T>  T max(vector<T>& a, ll n) { return *max_element(a.begin(), a.begin() + min(n, sz(a))); }
template<class T>  T max(vector<T>& a, ll s, ll n) { return *max_element(a.begin() + s, a.begin() + min(n, sz(a))); }
template<typename A, size_t N>  A max(A (&a)[N]) {    A res = a[0];    rep(i, N)res = max(res, a[i]);    return res;}template<typename A, size_t N, size_t O>  A max(A (&a)[N][O]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P>  A max(A (&a)[N][O][P]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q>  A max(A (&a)[N][O][P][Q], const T &v) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>  A max(A (&a)[N][O][P][Q][R]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S>  A max(A (&a)[N][O][P][Q][R][S]) {    A res = max(a[0]);    rep(i, N)res = max(res, max(a[i]));    return res;}
template<typename A, size_t N>  A min(A (&a)[N]) {    A res = a[0];    rep(i, N)res = min(res, a[i]);    return res;}template<typename A, size_t N, size_t O>  A min(A (&a)[N][O]) {    A res = min(a[0]);    rep(i, N)res = min(res, max(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P>  A min(A (&a)[N][O][P]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q>  A min(A (&a)[N][O][P][Q], const T &v) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R>  A min(A (&a)[N][O][P][Q][R]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}template<typename A, size_t N, size_t O, size_t P, size_t Q, size_t R, size_t S>  A min(A (&a)[N][O][P][Q][R][S]) {    A res = min(a[0]);    rep(i, N)res = min(res, min(a[i]));    return res;}
template<class T> T sum(vector<T> &v, ll s, ll t) {    T ret = 0;    rep(i, s, min(sz(v), t))ret += v[i];    return ret;}template<class T> T sum(vector<T> &v, ll t=inf) {    return sum(v, 0, t);}template<class T> T sum(vector<vector<T> > &v) {    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>  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(ll size) {    T *body = (T *) malloc((size * 2 + 1) * sizeof(T));    return body + size;}
template<class T> T *negarr2(ll h, ll w) {    double **dummy1 = new double *[2 * h + 1];    double *dummy2 = new double[(2 * h + 1) * (2 * w + 1)];    dummy1[0] = dummy2 + w;    for (ll i = 1; i <= 2 * h + 1; ++i) { dummy1[i] = dummy1[i - 1] + 2 * w + 1; }    double **a = dummy1 + h;    return a;}
//imoは0-indexed
//ruiは1-indexed
template<class T> vector<T> imo(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] += ret[i];    return ret;}
//kと同じものの数
template<class T, class U> vi imo(vector<T> &a, U k) {vector<T> ret = a;rep(i, sz(ret))ret[i] = a[i] == k;rep(i, sz(ret) - 1)ret[i + 1] += ret[i];return ret;}
template<class T> vector<T> imox(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)ret[i + 1] ^= ret[i];    return ret;}
//漸化的に最小を持つ
template<class T> vector<T> imi(vector<T> &v) {    vector<T> ret = v;    rep(i, sz(ret) - 1)chmi(ret[i + 1], ret[i]);    return ret;}
template<class T> struct ruiC {    const vector<T> rui;    ruiC(vector<T> &ru) : rui(ru) {}    T operator()(ll l, ll r) {        if (l > r) {            cerr<<"ruic ";deb(l, r);assert(0);        }        return rui[r] - rui[l];    }    T operator[](ll i) { return rui[i]; }    T back() { return rui.back(); }    ll size() { return rui.size(); }};
template<class T> struct rruic {    const T *rrui;    rruic(T *ru) : rrui(ru) {}     T operator()(ll l, ll r) {        assert(l >= r);        return rrui[r] - rrui[l];    }     T operator[](ll i) { return rrui[i]; }};
template<class T>ostream &operator<<(ostream &os, ruiC<T> a) {fora(v,a.rui)os<<v<<" ";return os;}
template<class T> vector<T> ruiv(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i];    return ret;}
template<class T> ruiC<T> ruic(vector<T> &a) {    vector<T> ret = ruiv(a);    return ruiC<T>(ret);}
vector<ll> ruiv(string &a) {    if (sz(a) == 0)return vi(1);    ll dec = ('0' <= a[0] && a[0] <= '9') ? '0' : 0;    vector<ll> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + a[i] - dec;    return ret;}
ruiC<ll> ruic(string &a) {    vector<ll> ret = ruiv(a);    return ruiC<ll>(ret);}
//kと同じものの数
template<class T, class U> vi ruiv(T &a, U k) {    vi ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] + (a[i] == k);    return ret;}
template<class T, class U> ruiC<ll> ruic(T &a, U k) {    vi ret = ruiv(a, k);    return ruiC<ll>(ret);}
//xor
template<class T> struct ruixC {    const vector<T> rui;    ruixC(vector<T> &ru) : rui(ru) {}    T operator()(ll l, ll r) {        if (l > r) {            cerr << "ruiXc ";            deb(l, r);            assert(0);        }        return rui[r] ^ rui[l];    }    T operator[](ll i) { return rui[i]; }    T back() { return rui.back(); }    ll size() { return rui.size(); }};
template<class T> vector<T> ruix(vector<T> &a) {    vector<T> ret(a.size() + 1);    rep(i, a.size())ret[i + 1] = ret[i] ^ a[i];    return ret;}
template<class T> ruixC<ll> ruixc(vector<T>  &a) {vi ret = ruix(a);return ruixC<ll>(ret);}
template<class T> vector<T> ruim(vector<T> &a) {    vector<T> res(a.size() + 1, 1);    rep(i, a.size())res[i + 1] = res[i] * a[i];    return res;}
//漸化的に最小を1indexで持つ
template<class T> vector<T> ruimi(vector<T> &a) {    ll n = sz(a);    vector<T> ret(n + 1);    rep(i, 1, n) {        ret[i] = a[i - 1];        chmi(ret[i + 1], ret[i]);    }    return ret;}
//template<class T> T *rrui(vector<T> &a) {
//右から左にかけての半開区間 (-1 n-1]
template<class T> rruic<T> rrui(vector<T> &a) {    ll len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    rer(i, len - 1)res[i - 1] = res[i] + a[i];    return rruic<T>(res);}
//掛け算
template<class T> T *rruim(vector<T> &a) {    ll len = a.size();    T *body = (T *) malloc((len + 1) * sizeof(T));    T *res = body + 1;    res[len - 1] = 1;    rer(i, len - 1)res[i - 1] = res[i] * a[i];    return res;}
template<class T, class U> 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); }
template<class T,class U> void dec(vector<T> &a,vector<U> &b) { for (auto &u :a)dec(u, 1);for (auto &u :b)dec(u, 1); }
template<class T,class U,class W> void dec(vector<T> &a,vector<U> &b,vector<W>&c ) { for (auto &u :a)dec(u, 1);for (auto &u :b)dec(u, 1);for (auto &u :c)dec(u, 1); }
bool ins(ll h, ll w, ll H, ll W) { return h >= 0 && w >= 0 && h < H && w < W; }
bool ins(ll l, ll v, ll r) { return l <= v && v < r; }
template<class T> bool ins(vector<T> &a, ll i, ll j = 0) { return ins(0, i, sz(a)) && ins(0, j, sz(a)); }
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()

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

char itoal(ll i) { return 'a' + i; }
char itoaL(ll i) { return 'A' + i; }
ll altoi(char c) {    if ('A' <= c && c <= 'Z')return c - 'A';    return c - 'a';}
ll ctoi(char c) { return c - '0'; }
char itoc(ll i) { return i + '0'; }
ll vtoi(vi &v) {    ll res = 0;    if (sz(v) > 18) {        debugline("vtoi");        deb(sz(v));        ole();    }    rep(i, sz(v)) {        res *= 10;        res += v[i];    }    return res;}
vi itov(ll i) {    vi res;    while (i) {        res.push_back(i % 10);        i /= 10;    }    rev(res);    return res;}
vi stov(string &a) {    ll n = sz(a);    vi ret(n);    rep(i, n) { ret[i] = a[i] - '0'; }    return ret;}
//基準を満たさないものは0になる
vi stov(string &a, char one) {    ll n = sz(a);    vi ret(n);    rep(i, n)ret[i] = a[i] == one;    return ret;}
vector<vector<ll>> ctoi(vector<vector<char>> s, char c) {    ll n = sz(s), m = sz(s[0]);    vector<vector<ll>> res(n, vector<ll>(m));    rep(i, n)rep(j, m)res[i][j] = s[i][j] == c;    return res;}
#define unique(v) v.erase( unique(v.begin(), v.end()), v.end() );
//[i] := i番として圧縮されたものを返す
vi compress(vi &a) {    vi b;    ll len = a.size();    for (ll i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) { a[i] = lower_bound(all(b), a[i]) - b.begin(); }    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, umap<ll, ll> &map) {    vi b;    ll len = a.size();    for (ll i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) {        ll v = a[i];        a[i] = lower_bound(all(b), a[i]) - b.begin();        map[v] = a[i];    }    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, vi &r) {    vi b;    ll len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (ll i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vi &a, vi &r, vi &s) {    vi b;    ll len = a.size();    fora(v, a)b.push_back(v);    fora(v, r)b.push_back(v);    fora(v, s)b.push_back(v);    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) a[i] = lower_bound(all(b), a[i]) - b.begin();    for (ll i = 0; i < sz(r); ++i) r[i] = lower_bound(all(b), r[i]) - b.begin();    for (ll i = 0; i < sz(s); ++i) r[i] = lower_bound(all(b), s[i]) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vector<vi> &a) {    vi b;    fora(vv, a)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vv, a)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
vi compress(vector<vector<vi >> &a) {    vi b;    fora(vvv, a)fora(vv, vvv)fora(v, vv)b.push_back(v);    sort(b);    unique(b);    fora(vvv, a)fora(vv, vvv)fora(v, vv)v = lower_bound(all(b), v) - b.begin();    ll blen = sz(b);    vi ret(blen);    rep(i, blen) { ret[i] = b[i]; }    return ret;}
void compress(ll a[], ll len) {    vi b;    for (ll i = 0; i < len; ++i) { b.push_back(a[i]); }    sort(b);    unique(b);    for (ll i = 0; i < len; ++i) { a[i] = lower_bound(all(b), a[i]) - b.begin(); }}
//要素が見つからなかったときに困る
#define binarySearch(a, v) (binary_search(all(a),v))
#define lowerIndex(a, v) (lower_bound(all(a),v)-a.begin())
#define lowerBound(a, v) (*lower_bound(all(a),v))
#define upperIndex(a, v) (upper_bound(all(a),v)-a.begin())
#define upperBound(a, v) (*upper_bound(all(a),v))
#define rlowerIndex(a, v) (upper_bound(all(a),v)-a.begin()-1)
#define rlowerBound(a, v) *(--(upper_bound(all(a),v)))
#define rupperIndex(a, v) (lower_bound(all(a),v)-a.begin()-1)
#define rupperBound(a, v) *(--(lower_bound(all(a),v)))
#define next2(a) next(next(a))
#define prev2(a) prev(prev(a))

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

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

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

//便利 数学 math
ll mod(ll a, ll m) { return (a % m + m) % m; }
ll pow(ll a) { return a * a; };
ll fact(ll v) { return v <= 1 ? 1 : v * fact(v - 1); }
dou factd(int v){static vd fact(2,1);    if(sz(fact)<=v){        rep(i,sz(fact),v+1){            fact.push_back(fact.back()*i);        }    }    return fact[v];}

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

ll gcd(ll a, ll b) {while (b) a %= b, swap(a, b);return abs(a);}
ll gcd(vi b) {ll res = b[0];rep(i, 1, sz(b))res = gcd(b[i], res);return res;}
ll lcm(ll a, ll b) { return a / gcd(a, b) * b; }
ll lcm(vi a) {ll res = a[0];rep(i, 1, sz(a))res = lcm(a[i], res);return res;}
ll ceil(ll a, ll b) {if (b == 0) {debugline("ceil");deb(a, b);ole();return -1;} else if (a < 0) { return 0; } else { return (a + b - 1) / b; }}
ll lower_remi__bx_a(ll kei, ll rem, ll x) {if (rem >= x) return 0;return (x - rem + kei - 1) / kei;}
ll lower_remv__bx_a(ll kei, ll rem, ll x) {if (rem >= x) return rem;return (x - rem + kei - 1) / kei * kei + rem;}
ll upper_remi__bx_a(ll kei, ll rem, ll x) {if (rem > x) return 0;return (x - rem + kei) / kei;}
ll upper_remv__bx_a(ll kei, ll rem, ll 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) >> 1; }
ll sig(ll s, ll t) { return ((s + t) * (t - s + 1)) >> 1; }


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

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

//@formatter:off
template<typename T> T minv(T a, T m);
template<typename T> T minv(T a);

template<typename T>
class Modular {
public:
    using Type = typename decay<decltype(T::value)>::type;    constexpr Modular() : value() {}    template<typename U>    Modular(const U &x) {        value = normalize(x);    }    template<typename U>    static Type normalize(const U &x) {        Type v;        if (-mod() <= x && x < mod()) v = static_cast<Type>(x);        else v = static_cast<Type>(x % mod());        if (v < 0) v += mod();        return v;    }    const Type &operator()() const { return value; }    template<typename U>explicit operator U() const { return static_cast<U>(value); }    constexpr static Type mod() { return T::value; }    Modular &operator+=(const Modular &other) {        if ((value += other.value) >= mod()) value -= mod();        return *this;    }    Modular &operator-=(const Modular &other) {        if ((value -= other.value) < 0) value += mod();        return *this;    }    template<typename U> Modular &operator+=(const U &other) { return *this += Modular(other); }    template<typename U> Modular &operator-=(const U &other) { return *this -= Modular(other); }    Modular &operator++() { return *this += 1; }    Modular &operator--() { return *this -= 1; }    Modular operator++(signed) {        Modular result(*this);        *this += 1;        return result;    }    Modular operator--(signed) {        Modular result(*this);        *this -= 1;        return result;    }    Modular operator-() const { return Modular(-value); }
    template<typename U = T>typename enable_if<is_same<typename Modular<U>::Type, signed>::value, Modular>::type &operator*=(const Modular &rhs) {
#ifdef _WIN32
        uint64_t x = static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value);uint32_t xh = static_cast<uint32_t>(x >> 32), xl = static_cast<uint32_t>(x), d, m;asm("divl %4; \n\t": "=a" (d), "=d" (m): "d" (xh), "a" (xl), "r" (mod()));value = m;
#else
        value = normalize(static_cast<int64_t>(value) * static_cast<int64_t>(rhs.value));
#endif
        return *this;
    }
    template<typename U = T>    typename enable_if<is_same<typename Modular<U>::Type, int64_t>::value, Modular>::type &operator*=(const Modular &rhs) {        int64_t q = static_cast<int64_t>(static_cast<double>(value) * rhs.value / mod());        value = normalize(value * rhs.value - q * mod());        return *this;    }    template<typename U = T>    typename enable_if<!is_integral<typename Modular<U>::Type>::value, Modular>::type &operator*=(const Modular &rhs) {        value = normalize(value * rhs.value);        return *this;    }    Modular &operator/=(const Modular &other) { return *this *= Modular(minv(other.value)); }    template<typename U> friend bool operator==(const Modular<U> &lhs, const Modular<U> &rhs);    template<typename U> friend bool operator<(const Modular<U> &lhs, const Modular<U> &rhs);    template<typename U> friend std::istream &operator>>(std::istream &stream, Modular<U> &number);    operator int() { return value; }private:    Type value;
};
template<typename T> bool operator==(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value == rhs.value; }template<typename T, typename U> bool operator==(const Modular<T> &lhs, U rhs) { return lhs == Modular<T>(rhs); }template<typename T, typename U> bool operator==(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) == rhs; }template<typename T> bool operator!=(const Modular<T> &lhs, const Modular<T> &rhs) { return !(lhs == rhs); }template<typename T, typename U> bool operator!=(const Modular<T> &lhs, U rhs) { return !(lhs == rhs); }template<typename T, typename U> bool operator!=(U lhs, const Modular<T> &rhs) { return !(lhs == rhs); }template<typename T> bool operator<(const Modular<T> &lhs, const Modular<T> &rhs) { return lhs.value < rhs.value; }template<typename T> Modular<T> operator+(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }template<typename T, typename U> Modular<T> operator+(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) += rhs; }template<typename T, typename U> Modular<T> operator+(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) += rhs; }template<typename T> Modular<T> operator-(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }template<typename T, typename U> Modular<T> operator-(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) -= rhs; }template<typename T, typename U> Modular<T> operator-(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) -= rhs; }template<typename T> Modular<T> operator*(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }template<typename T, typename U> Modular<T> operator*(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) *= rhs; }template<typename T, typename U> Modular<T> operator*(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) *= rhs; }template<typename T> Modular<T> operator/(const Modular<T> &lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }template<typename T, typename U> Modular<T> operator/(const Modular<T> &lhs, U rhs) { return Modular<T>(lhs) /= rhs; }template<typename T, typename U> Modular<T> operator/(U lhs, const Modular<T> &rhs) { return Modular<T>(lhs) /= rhs; }

constexpr signed MOD =
//        998244353;
1e9 + 7;//MOD
using mint = Modular<std::integral_constant<decay<decltype(MOD)>::type, MOD>>;
constexpr int mint_len = 1400001;
vi fac, finv, inv;
vi p2;
mint com(int n, int r) {    if (r < 0 || r > n) return 0;    return mint(finv[r] * fac[n] % MOD * finv[n - r]);}
mint pom(int n, int r) {/*    if (!sz(fac)) com(0, -1);*/    if (r < 0 || r > n) return 0;    return mint(fac[n] * finv[n - 1]);}
mint npr(int n, int r) {/*    if (!sz(fac)) com(0, -1);*/    if (r < 0 || r > n) return 0;    return mint(fac[n] * finv[n - r]);}
int nprin(int n, int r) {/*    if (!sz(fac)) com(0, -1);*/    if (r < 0 || r > n) return 0;    return fac[n] * finv[n - r] % MOD;}
int icom(int n, int r) {    const int NUM_ = 1400001;    static ll fac[NUM_ + 1], finv[NUM_ + 1], inv[NUM_ + 1];    if (fac[0] == 0) {        inv[1] = fac[0] = finv[0] = 1;        for (int i = 2; i <= NUM_; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;        for (int i = 1; i <= NUM_; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;    }    if (r < 0 || r > n) return 0;    return ((finv[r] * fac[n] % MOD) * finv[n - r]) % MOD;}
#define ncr com
#define ncri icom
//n個の場所にr個の物を置く
mint nhr(int n, int r) { return com(n + r - 1, r); }
mint hom(int n, int r) { return com(n + r - 1, r); }
int nhri(int n, int r) { return icom(n + r - 1, r); }
template<typename T> T minv(T a, T m) {    T u = 0, v = 1;    while (a != 0) {        T t = m / a;        m -= t * a;        swap(a, m);        u -= t * v;        swap(u, v);    }    assert(m == 1);    return u;}
template<typename T> T minv(T a) {    if (a < mint_len)return inv[a];    T u = 0, v = 1;    T m = MOD;    while (a != 0) {        T t = m / a;        m -= t * a;        swap(a, m);        u -= t * v;        swap(u, v);    }    assert(m == 1);    return u;}
template<typename T, typename U> Modular<T> mpow(const Modular<T> &a, const U &b) {    assert(b >= 0);    int x = a(), res = 1;    U p = b;    while (p > 0) {        if (p & 1) (res *= x) %= MOD;        (x *= x) %= MOD;        p >>= 1;    }    return res;}
template<typename T, typename U, typename V> mint mpow(const T a, const U b, const V m = MOD) {    assert(b >= 0);    int x = a, res = 1;    U p = b;    while (p > 0) {        if (p & 1) (res *= x) %= m;        (x *= x) %= m;        p >>= 1;    }    return res;}
template<typename T, typename U> mint mpow(const T a, const U b) {    assert(b >= 0);    int x = a, res = 1;    U p = b;    while (p > 0) {        if (p & 1) (res *= x) %= MOD;        (x *= x) %= MOD;        p >>= 1;    }    return res;}
template<typename T, typename U, typename V> int mpowi(const T &a, const U &b, const V &m = MOD) {    assert(b >= 0);    int x = a, res = 1;    U p = b;    while (p > 0) {        if (p & 1) (res *= x) %= m;        (x *= x) %= m;        p >>= 1;    }    return res;}
template<typename T> string to_string(const Modular<T> &number) {    return to_string(number());}
string yuri(const mint &a) {    stringstream st;    rep(i, 300) {rep(j, 300) {if ((mint) i / j == a) {st << i << " / " << j;return st.str();}}}    rep(i, 1000) {rep(j, 1000) {if ((mint) i / j == a) {st << i << " / " << j;return st.str();}}}    return st.str();}
template<typename T> std::ostream &operator<<(std::ostream &stream, const Modular<T> &number) {stream << number();
#ifdef _DEBUG
//    stream << " -> " << yuri(number);
#endif
    return stream;
}
//@formatter:off
template<typename T> std::istream &operator>>(std::istream &stream, Modular<T> &number) {    typename common_type<typename Modular<T>::Type, int64_t>::type x;    stream >> x;    number.value = Modular<T>::normalize(x);    return stream;}
using PM = pair<mint, mint>;
using vm = vector<mint>;
using mapm = map<int, mint>;
using umapm = umap<int, mint>;
#define vvm(...) o_vvt(__VA_ARGS__,vvt4,vvt3,vvt2 ,vvt1,vvt0)(mint,__VA_ARGS__)
#define vnm(name, ...) auto name = make_v<mint>(__VA_ARGS__)

struct setmod{
    setmod() {
//    p2.resize(mint_len);p2[0] = 1; for (int i = 1; i < mint_len; ++i) p2[i] = p2[i - 1] * 2 % MOD;
        fac.resize(mint_len);    finv.resize(mint_len);    inv.resize(mint_len);    inv[1] = fac[0] = finv[0] = 1;    for (int i = 2; i < mint_len; ++i) inv[i] = inv[MOD % i] * (MOD - MOD / i) % MOD;    for (int i = 1; i < mint_len; ++i) fac[i] = fac[i - 1] * i % MOD, finv[i] = finv[i - 1] * inv[i] % MOD;
    }
}setmodv;
//@formatter:on
//nhr n個の場所にr個の物を分ける

//@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 mll pr
//上下左右
const string udlr = "udlr";
string UDLR = "UDLR";//x4と連動 UDLR.find('U') := x4[0]
//右、上が正
constexpr ll y4[] = {1, -1, 0, 0};
constexpr ll x4[] = {0, 0, -1, 1};
constexpr ll y8[] = {0, 1, 0, -1, -1, 1, 1, -1};
constexpr ll x8[] = {1, 0, -1, 0, 1, -1, 1, -1};


ll n, m, k, K, d, H, W, x, y, z, q, Q;
ll cou;
vi a, b, c;
//vvi (s, 0, 0);
vvc (ba, 0, 0);
vp p;
str s;

//幾何ライブラリ
struct V {/*@formatter:off*/
    double x, y;
    V(double x=0, double y=0): x(x), y(y) {}
    V& operator+=(const V& v) { x += v.x; y += v.y; return *this;}
    V operator+(const V& v) const { return V(*this) += v;}
    V& operator-=(const V& v) { x -= v.x; y -= v.y; return *this;}
    V operator-(const V& v) const { return V(*this) -= v;}
    V& operator*=(double s) { x *= s; y *= s; return *this;}
    V operator*(double s) const { return V(*this) *= s;}
    V& operator/=(double s) { x /= s; y /= s; return *this;}
    V operator/(double s) const { return V(*this) /= s;}
    //内積
    double dot(const V& v) const { return x*v.x + y*v.y;}
    //外積
    double cross(const V& v) const { return x*v.y - v.x*y;}
    double norm2() const { return x*x + y*y;}
    double norm() const { return sqrt(norm2());}
    // 象限
    int ort() const {        if (abs(x) < eps && abs(y) < eps) return 0;        if (y > 0) return x>0 ? 1 : 2;        else return x<0 ? 3 : 4;    }
    bool operator<(const V& v) const {        int o = ort(), vo = v.ort();        if (o != vo) return o < vo;        return cross(v) > 0;    }
};
istream& operator>>(istream& ist, V& v) {ist >> v.x >> v.y; return ist;}
ostream& operator<<(ostream& os, const V& v) {os<<"("<<v.x<<","<<v.y<<")"; return os;}
using dots = vector<V>;

double cross(const V &p, const V &q) { return p.x * q.y - p.y * q.x; }
double dis(const V &p, const V &q){return std::sqrt(pow(p.x-q.x,2) +pow(p.y-q.y,2)) ;}
//c-a, a-bとした時のaの角度
/*@formatter:on*/
//偏角
double arg(V &a) { return atan2(a.y, a.x); }
//角度
double ang(V c, V a, V b) {
    c -= a;
    b -= a;
    auto e = arg(c);
    auto f = arg(b);
    double ret = abs(arg(c) - arg(b));
    return min(ret, 2 * PI - ret);
}
/*@formatter:off*/
// 一直線上の3点を含めない
// 全ベクトルが同じ方向でどれを足し合わせるかを考えるときに一直線にすると爆発するhttps://atcoder.jp/contests/abc139/tasks/abc139_f F-Engines
//反時計回り
vector<V> convexhull(vector<V> &ps) {int n = (int) ps.size();    if (n <= 2) {        return ps;    }    vector<V> res(2 * n);    std::sort(ps.begin(), ps.end(), [&](V &l, V &r) {        if (l.x != r.x)return l.x < r.x;        else return l.y < r.y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < eps) {                --k;                if (k < 2) break;            }        }        res[k] = ps[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < eps) {                --k;                if (k < t) break;            }        }        res[k] = ps[i];        ++k;    }    res.resize(k - 1);    return res;}

//インデックスを返す
vi convexhull_i(vector<V> &ps) {    int n = (int) ps.size();    if (n <= 2) {        if (n == 1)return vi{0};else return vi{0, 1};}vi res(2 * n);vi ind = iota(0, n);    std::sort(ind.begin(), ind.end(), [&](int l, int r) {        if (ps[l].x != ps[r].x)return ps[l].x < ps[r].x;        else return ps[l].y < ps[r].y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < eps) {                --k;                if (k < 2) break;            }        }        res[k] = ind[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < eps) {                --k;                if (k < t) break;            }        }        res[k] = ind[i];        ++k;    }    res.resize(k - 1);    return res;}

// 一直線上の3点を含める
vector<V> convexhull_line(vector<V> &ps) {    int n = (int) ps.size();    if (n <= 2) { return ps; }    vector<V> res(2 * n);    std::sort(ps.begin(), ps.end(), [&](V &l, V &r) { if (l.x != r.x)return l.x < r.x; else return l.y < r.y; });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < -eps) {                --k;                if (k < 2) break;            }        }        res[k] = ps[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(res[k - 1] - res[k - 2], ps[i] - res[k - 2]) < -eps) {                --k;                if (k < t) break;            }        }        res[k] = ps[i];        ++k;    }    res.resize(k - 1);    return res;}

vi convexhull_line_i(vector<V> &ps) {    int n = (int) ps.size();    if (n <= 2) {        if (n == 1)return vi{0};        else return vi{0, 1};    }    vi res(2 * n);    vi ind = iota(0, n);    std::sort(ind.begin(), ind.end(), [&](int l, int r) {        if (ps[l].x != ps[r].x)return ps[l].x < ps[r].x;        else return ps[l].y < ps[r].y;    });    int k = 0;    for (int i = 0; i < n; ++i) {        if (k >= 2) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < -eps) {                --k;                if (k < 2) break;            }        }        res[k] = ind[i];        ++k;    }    int t = k + 1;    for (int i = n - 2; i >= 0; --i) {        if (k >= t) {            while (cross(ps[res[k - 1]] - ps[res[k - 2]], ps[ind[i]] - ps[res[k - 2]]) < -eps) {                --k;                if (k < t) break;            }        }        res[k] = ind[i];        ++k;    }    res.resize(k - 1);    return res;}/*@formatter:on*/

void solve() {
    in(n);
    dots d;
    na(d, n);
    vi c = convexhull_i(d);
    vd res(n);
    foriv(i, v, c) {
        dou rad = ang(d[prev], d[v], d[next]);
        res[v] = (PI - rad) / (2 * PI);
    }
    outl(res);
}

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

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

signed main() {
    solve();

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


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


Measured memory_bytes: 33152