結果

問題 No.2808 Concentration
ユーザー hamamu
提出日時 2024-07-12 23:10:10
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 219 ms / 2,000 ms
コード長 60,039 bytes
コンパイル時間 7,949 ms
コンパイル使用メモリ 299,364 KB
最終ジャッジ日時 2025-02-22 05:19:59
ジャッジサーバーID
（参考情報） 		judge2 / judge3
このコードへのチャレンジ
（要ログイン）
ファイルパターン 結果
other AC * 57
権限があれば一括ダウンロードができます

ソースコード

diff #
プレゼンテーションモードにする

#if !defined(MYLOCAL)//提出時用テンプレート
#pragma GCC optimize("Ofast")
#if defined(NDEBUG)
#undef NDEBUG
#endif
#include "bits/stdc++.h"
#if !defined(_MSC_VER) && __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
#endif
using namespace std;
using     ll=long long;     using   dd=double;     
using    vll=vector<   ll>; using  vdd=vector< dd>;
using   vvll=vector<  vll>; using vvdd=vector<vdd>;
using  vvvll=vector< vvll>; using vvvdd=vector<vvdd>;
using vvvvll=vector<vvvll>;
using   pll=pair<ll,ll>;  using   tll=tuple<ll,ll,ll>; using   qll=tuple<ll,ll,ll,ll>;
using  vpll=vector< pll>; using  vtll=vector< tll>;    using  vqll=vector< qll>;
using vvpll=vector<vpll>; using vvtll=vector<vtll>;    using vvqll=vector<vqll>;
using namespace chrono;
constexpr ll INF = 1001001001001001001;
struct Fast{ Fast(){ cin.tie(0); ios::sync_with_stdio(false); cout<<fixed<<setprecision(numeric_limits<double>::max_digits10); } } fast;
#define REPS(i, S, E) for (ll i = (S); i <= (E); i++)
#define REP(i, N) REPS(i, 0, (N)-1)
#define DEPS(i, S, E) for (ll i = (E); i >= (S); i--)
#define DEP(i, N) DEPS(i, 0, (N)-1)
#define EXPAND( x ) x//VS用おまじない
#define overload3(_1,_2,_3,name,...) name
#define overload4(_1,_2,_3,_4,name,...) name
#define overload5(_1,_2,_3,_4,_5,name,...) name
#define rep3(i, S, E)    for (ll i = (S); i <= (E); i++)
#define rep4(i, S, E, t) for (ll i = (S); i <= (E); i+=(t))
#define rep(...) EXPAND(overload4(__VA_ARGS__,rep4,rep3,_,_)(__VA_ARGS__))
#define dep3(i, E, S)    for (ll i = (E); i >= (S); i--)
#define dep4(i, E, S, t) for (ll i = (E); i >= (S); i-=(t))
#define dep(...) EXPAND(overload4(__VA_ARGS__, dep4, dep3,_,_)(__VA_ARGS__))
#define each2(e,v) for (auto && e:v)
#define each3(a,b,v) for (auto &&[a,b]:v)
#define each4(a,b,c,v) for (auto &&[a,b,c]:v)
#define each5(a,b,c,d,v) for (auto &&[a,b,c,d]:v)
#define each(...) EXPAND(overload5(__VA_ARGS__,each5,each4,each3,each2,_)(__VA_ARGS__))
#define ALL1(v)     (v).begin(),     (v).end()
#define ALL2(v,E)   (v).begin(),     (v).begin()+((E)+1)
#define ALL3(v,S,E) (v).begin()+(S), (v).begin()+((E)+1)
#define ALL(...) EXPAND(overload3(__VA_ARGS__, ALL3, ALL2, ALL1)(__VA_ARGS__))
#define all ALL
#define RALL1(v)     (v).rbegin(),     (v).rend()
#define RALL2(v,E)   (v).rbegin(),     (v).rbegin()+((E)+1)
#define RALL3(v,S,E) (v).rbegin()+(S), (v).rbegin()+((E)+1)
#define RALL(...) EXPAND(overload3(__VA_ARGS__, RALL3, RALL2, RALL1)(__VA_ARGS__))
#define rall RALL
template<class T> inline bool chmax(T &a, T b) { if (a < b) { a = b; return true; }return false; }
template<class T> inline bool chmin(T &a, T b) { if (a > b) { a = b; return true; }return false; }
template<class T> inline T MaxE(vector<T>&v,ll S,ll E){ T m=v[S]; rep(i,S,E)chmax(m,v[i]); return m; }
template<class T> inline T MinE(vector<T>&v,ll S,ll E){ T m=v[S]; rep(i,S,E)chmin(m,v[i]); return m; }
template<class T> inline T MaxE(vector<T> &v) { return MaxE(v,0,(ll)v.size()-1); }
template<class T> inline T MinE(vector<T> &v) { return MinE(v,0,(ll)v.size()-1); }
template<class T> inline auto maxe(T &&v,ll S,ll E){ return *max_element(ALL(v,S,E)); }
template<class T> inline auto maxe(T &&v){ return *max_element(ALL(v)); }
template<class T> inline auto mine(T &&v,ll S,ll E){ return *min_element(ALL(v,S,E)); }
template<class T> inline auto mine(T &&v){ return *min_element(ALL(v)); }
template<class T> inline T Sum(vector<T> &v,ll S,ll E){ T s=T(); rep(i,S,E)s+=v[i]; return s; }
template<class T> inline T Sum(vector<T> &v) { return Sum(v,0,v.size()-1); }
template<class T,class U=typename remove_reference<T>::type::value_type>
inline U sum(T &&v,ll S,ll E) {return accumulate(all(v,S,E),U());}
template<class T> inline auto sum(T &&v) {return sum(v,0,v.end()-v.begin()-1);}
template<class T> inline ll sz(T &&v){ return (ll)v.size(); }
inline ll CEIL(ll a,ll b){ return (a<0) ? -(-a/b) : (a+b-1)/b; } //負もOK
inline ll FLOOR(ll a,ll b){ return -CEIL(-a,b); } //負もOK
//pair用テンプレート
template<class T,class S> inline pair<T,S>& operator+=(pair<T,S> &a,const pair<T,S> &b){ a.first+=b.first; a.second+=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator-=(pair<T,S> &a,const pair<T,S> &b){ a.first-=b.first; a.second-=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator*=(pair<T,S> &a,const pair<T,S> &b){ a.first*=b.first; a.second*=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator/=(pair<T,S> &a,const pair<T,S> &b){ a.first/=b.first; a.second/=b.second; return a; }
template<class T,class S> inline pair<T,S>& operator%=(pair<T,S> &a,const pair<T,S> &b){ a.first%=b.first; a.second%=b.second; return a; }
template<class T,class S,class R> inline pair<T,S>& operator+=(pair<T,S> &a,R b){ a.first+=b; a.second+=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator-=(pair<T,S> &a,R b){ a.first-=b; a.second-=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator*=(pair<T,S> &a,R b){ a.first*=b; a.second*=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator/=(pair<T,S> &a,R b){ a.first/=b; a.second/=b; return a; }
template<class T,class S,class R> inline pair<T,S>& operator%=(pair<T,S> &a,R b){ a.first%=b; a.second%=b; return a; }
template<class T,class S,class R> inline pair<T,S> operator+(const pair<T,S> &a,R b){ pair<T,S> c=a; return c+=b; }
template<class T,class S,class R> inline pair<T,S> operator-(const pair<T,S> &a,R b){ pair<T,S> c=a; return c-=b; }
template<class T,class S,class R> inline pair<T,S> operator*(const pair<T,S> &a,R b){ pair<T,S> c=a; return c*=b; }
template<class T,class S,class R> inline pair<T,S> operator/(const pair<T,S> &a,R b){ pair<T,S> c=a; return c/=b; }
template<class T,class S,class R> inline pair<T,S> operator%(const pair<T,S> &a,R b){ pair<T,S> c=a; return c%=b; }
template<class T,class S,class R> inline pair<T,S> operator-(R b,const pair<T,S> &a){ pair<T,S> c=-a; return c+=b; }
template<class T,class S> inline pair<T,S> operator-(const pair<T,S> &a,const pair<T,S> &b){ pair<T,S> c=a; return c-=b; }
template<class T,class S> inline pair<T,S> operator-(const pair<T,S> &a){ pair<T,S> c=a; return c*=(-1); }
template<class T,class S> inline ostream &operator<<(ostream &os,const pair<T,S> &a){ return os << a.first << ' ' << a.second; }
//tuple用テンプレート 出力用のみ
template<class T,class S,class R> inline ostream &operator<<(ostream &os,const tuple<T,S,R> &a){ return os << get<0>(a) << ' ' << get<1>(a) << ' ' 
    << get<2>(a); }
template<class T,class S,class R,class Q> inline ostream &operator<<(ostream &os,const tuple<T,S,R,Q> &a){ return os << get<0>(a) << ' ' << get<1>(a
    ) << ' ' << get<2>(a) << ' ' << get<3>(a); }
//vector用テンプレート
template<class T> inline ostream &operator<<(ostream &os,const vector<T> &a){ for (ll i=0; i<(ll)a.size(); i++) os<<(i>0?" ":"")<<a[i];  return os; 
    }
//array用テンプレート
template<class T,size_t S> inline array<T,S>& operator+=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]+=b[i];  return a; }
template<class T,size_t S> inline array<T,S>& operator-=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]-=b[i];  return a; }
template<class T,size_t S> inline array<T,S>& operator*=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]*=b[i];  return a; }
template<class T,size_t S> inline array<T,S>& operator/=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]/=b[i];  return a; }
template<class T,size_t S> inline array<T,S>& operator%=(array<T,S> &a,const array<T,S> &b){ for (ll i=0; i<(ll)S; i++) a[i]%=b[i];  return a; }
template<class T,size_t S,class R> inline array<T,S>& operator+=(array<T,S> &a,R b){ for (T &e: a) e+=b;  return a; }
template<class T,size_t S,class R> inline array<T,S>& operator-=(array<T,S> &a,R b){ for (T &e: a) e-=b;  return a; }
template<class T,size_t S,class R> inline array<T,S>& operator*=(array<T,S> &a,R b){ for (T &e: a) e*=b;  return a; }
template<class T,size_t S,class R> inline array<T,S>& operator/=(array<T,S> &a,R b){ for (T &e: a) e/=b;  return a; }
template<class T,size_t S,class R> inline array<T,S>& operator%=(array<T,S> &a,R b){ for (T &e: a) e%=b;  return a; }
template<class T,size_t S,class R> inline array<T,S> operator+(const array<T,S> &a,R b){ array<T,S> c=a; return c+=b; }
template<class T,size_t S,class R> inline array<T,S> operator-(const array<T,S> &a,R b){ array<T,S> c=a; return c-=b; }
template<class T,size_t S,class R> inline array<T,S> operator*(const array<T,S> &a,R b){ array<T,S> c=a; return c*=b; }
template<class T,size_t S,class R> inline array<T,S> operator/(const array<T,S> &a,R b){ array<T,S> c=a; return c/=b; }
template<class T,size_t S,class R> inline array<T,S> operator%(const array<T,S> &a,R b){ array<T,S> c=a; return c%=b; }
template<class T,size_t S,class R> inline array<T,S> operator-(R b,const array<T,S> &a){ array<T,S> c=-a; return c+=b; }
template<class T,size_t S> inline array<T,S> operator-(const array<T,S> &a,const array<T,S> &b){ array<T,S> c=a; return c-=b; }
template<class T,size_t S> inline array<T,S> operator-(const array<T,S> &a){ array<T,S> c=a; return c*=(-1); }
template<class T,size_t S> inline ostream &operator<<(ostream &os,const array<T,S> &a){ for (ll i=0; i<(ll)S; i++) os<<(i>0?" ":"")<<a[i];  return 
    os; }
template<class T,size_t S,class R> struct view1d;
template<class T,size_t S,class R> struct view1dIter{
    view1d<T,S,R> *vw=nullptr;
    ll idx=INF;
    view1dIter(){}
    view1dIter(view1d<T,S,R> *vw_,ll idx_):vw(vw_),idx(idx_){}
    view1dIter(const view1dIter<T,S,R> &it):vw(it.vw),idx(it.idx){}
    R &operator*()          { return (*vw)[idx]; }
    R &operator*()     const{ return (*vw)[idx]; }
    R &operator[](ll i)     { return (*vw)[idx+i]; }
    R &operator[](ll i)const{ return (*vw)[idx+i]; }
    auto &operator++(){ idx++; return *this; }
    auto &operator--(){ idx--; return *this; }
    auto operator++(int){ auto it=*this; idx++; return it; }
    auto operator--(int){ auto it=*this; idx--; return it; }
    auto &operator+=(ll n){ idx+=n; return *this; }
    auto &operator-=(ll n){ idx-=n; return *this; }
    auto operator+(ll n){ auto it=*this; return it+=n; }
    auto operator-(ll n){ auto it=*this; return it-=n; }
    ll   operator- (const view1dIter<T,S,R> &it)const{ return idx- it.idx; }
    bool operator< (const view1dIter<T,S,R> &it)const{ return idx< it.idx; }
    bool operator> (const view1dIter<T,S,R> &it)const{ return idx> it.idx; }
    bool operator<=(const view1dIter<T,S,R> &it)const{ return idx<=it.idx; }
    bool operator>=(const view1dIter<T,S,R> &it)const{ return idx>=it.idx; }
    bool operator!=(const view1dIter<T,S,R> &it)const{ return idx!=it.idx; }
    bool operator==(const view1dIter<T,S,R> &it)const{ return idx==it.idx; }
    using iterator_category = random_access_iterator_tag;
    using value_type = R;
    using difference_type = ll;
    using pointer = R*;
    using reference = R&;
};
template<class T,size_t S,class R> struct view1d{
    using Sll=array<ll,S>;
    T &data; //参照先データ
    Sll dsize; //参照先データ各軸のサイズ
    Sll s=Sll(); //view先頭
    Sll d=Sll(); //view方向
    ll len; //view長さ
    R &dummy; //範囲外の値参照
    R dummyj=R(); //範囲外の値実体
    /*---- コンストラクタ ----*/
    view1d(T &v): data(v),dummy(dummyj){
        SetDsize(v,dsize);
        d[S-1]=1;
        len=dsize[S-1];
    }
    //このコンストラクタは、マイナス座標指定を末尾にしない(暫定)
    //代入時Array<ll,2>{...}等と書くため、使い勝手悪い。将来廃止するかも。
    view1d(T &v,Sll s,Sll e,Sll d): data(v),s(s),d(d),len(Len(s,e,d)),dummy(dummyj){
        SetDsize(v,dsize);
    }
    view1d(T &v,Sll s,Sll d,ll len,R &dmy_): data(v),s(s),d(d),len(len),dummy(dmy_){
        SetDsize(v,dsize);
    }
    /*---- 演算 ----*/
    template<class Q> auto &operator =(const Q &b){ rep(i,0,len-1) (*this)[i] =b; return *this; }
    template<class Q> auto &operator+=(const Q &b){ rep(i,0,len-1) (*this)[i]+=b; return *this; }
    template<class Q> auto &operator-=(const Q &b){ rep(i,0,len-1) (*this)[i]-=b; return *this; }
    template<class Q> auto &operator*=(const Q &b){ rep(i,0,len-1) (*this)[i]*=b; return *this; }
    template<class Q> auto &operator/=(const Q &b){ rep(i,0,len-1) (*this)[i]/=b; return *this; }
    template<class Q> auto &operator%=(const Q &b){ rep(i,0,len-1) (*this)[i]%=b; return *this; }
    auto &operator =(const string &b){ return CpSeq(b); }
    template<size_t Q> auto &operator =(const char (&b)[Q]){ return *this=string(b); }
    template<class Q> auto &operator =(const vector<Q> &b){ return CpSeq(b); }
    template<class Q> auto &operator+=(const vector<Q> &b){ return PlSeq(b); }
    template<class Q> auto &operator-=(const vector<Q> &b){ return MnSeq(b); }
    template<class Q> auto &operator*=(const vector<Q> &b){ return PrSeq(b); }
    template<class Q> auto &operator/=(const vector<Q> &b){ return DvSeq(b); }
    template<class Q> auto &operator%=(const vector<Q> &b){ return RmSeq(b); }
    template<class Q,size_t P,class O> auto &operator =(const view1d<Q,P,O> &b){ return CpSeq(b); }
    template<class Q,size_t P,class O> auto &operator+=(const view1d<Q,P,O> &b){ return PlSeq(b); }
    template<class Q,size_t P,class O> auto &operator-=(const view1d<Q,P,O> &b){ return MnSeq(b); }
    template<class Q,size_t P,class O> auto &operator*=(const view1d<Q,P,O> &b){ return PrSeq(b); }
    template<class Q,size_t P,class O> auto &operator/=(const view1d<Q,P,O> &b){ return DvSeq(b); }
    template<class Q,size_t P,class O> auto &operator%=(const view1d<Q,P,O> &b){ return RmSeq(b); }
    template<class Q> auto &CpSeq(const Q &b){ rep(i,0,min(sz(b),len)-1) (*this)[i] =b[i]; return *this; }
    template<class Q> auto &PlSeq(const Q &b){ rep(i,0,min(sz(b),len)-1) (*this)[i]+=b[i]; return *this; }
    template<class Q> auto &MnSeq(const Q &b){ rep(i,0,min(sz(b),len)-1) (*this)[i]-=b[i]; return *this; }
    template<class Q> auto &PrSeq(const Q &b){ rep(i,0,min(sz(b),len)-1) (*this)[i]*=b[i]; return *this; }
    template<class Q> auto &DvSeq(const Q &b){ rep(i,0,min(sz(b),len)-1) (*this)[i]/=b[i]; return *this; }
    template<class Q> auto &RmSeq(const Q &b){ rep(i,0,min(sz(b),len)-1) (*this)[i]%=b[i]; return *this; }
    //template<class Q,class P> static bool eq(const Q &a,const P &b){
    //  return equals(ALL(a),ALL(b));
    //}
    template<class Q,class P> static bool eq(const Q &a,const P &b){
        if ((ll)a.size()!=(ll)b.size()) return false;
        rep(i,0,(ll)a.size()-1){
            if (a[i]!=b[i]) return false;
        }
        return true;
    }
    template<class Q,class P> static bool lt(const Q &a,const P &b){
        ll n=min((ll)a.size(),(ll)b.size());
        rep(i,0,n-1){
            if (a[i]<b[i])return true;
            if (a[i]>b[i])return false;
        }
        return (ll)a.size()<(ll)b.size();
    }
    template<class Q,size_t P,class O> bool operator==(           const view1d<Q,P,O> &b){ return eq(*this,b); }
    template<class Q>                  bool operator==(           const             Q &b){ return eq(*this,b); }
    template<class Q> friend           bool operator==(const Q &a,const view1d<T,S,R> &b){ return eq(    a,b); }
    template<size_t Q>                 bool operator==(           const     char (&b)[Q]){ return eq(*this,string(b)); }
    template<class Q,size_t P,class O> bool operator!=(           const view1d<Q,P,O> &b){ return !(*this==b); }
    template<class Q>                  bool operator!=(           const             Q &b){ return !(*this==b); }
    template<class Q> friend           bool operator!=(const Q &a,const view1d<T,S,R> &b){ return !(    a==b); }
    template<class Q,size_t P,class O> bool operator< (           const view1d<Q,P,O> &b){ return lt(*this,b); }
    template<class Q>                  bool operator< (           const             Q &b){ return lt(*this,b); }
    template<class Q> friend           bool operator< (const Q &a,const view1d<T,S,R> &b){ return lt(    a,b); }
    template<class Q,size_t P,class O> bool operator> (           const view1d<Q,P,O> &b){ return lt(b,*this); }
    template<class Q>                  bool operator> (           const             Q &b){ return lt(b,*this); }
    template<class Q> friend           bool operator> (const Q &a,const view1d<T,S,R> &b){ return lt(b,a    ); }
    template<class Q,size_t P,class O> bool operator<=(           const view1d<Q,P,O> &b){ return !(*this>b); }
    template<class Q>                  bool operator<=(           const             Q &b){ return !(*this>b); }
    template<class Q> friend           bool operator<=(const Q &a,const view1d<T,S,R> &b){ return !(    a>b); }
    template<class Q,size_t P,class O> bool operator>=(           const view1d<Q,P,O> &b){ return !(*this<b); }
    template<class Q>                  bool operator>=(           const             Q &b){ return !(*this<b); }
    template<class Q> friend           bool operator>=(const Q &a,const view1d<T,S,R> &b){ return !(    a<b); }
    /*---- getter ----*/
    ll size() const { return len; }
    R &operator[](ll i){ return const_cast<R&>(OrgAt(s+d*i)); }
    const R &operator[](ll i) const { return OrgAt(s+d*i); }
    R &at(ll i){ if(i<0)i+=len; return (*this)[i]; }
    //vector<R> tov(){ vector<R> vvv(len);  rep(i,0,len-1) vvv[i]=(*this)[i];  return vvv; }
    operator vector<R>(){ vector<R> v(len);  rep(i,0,len-1) v[i]=(*this)[i];  return v; }
    bool contains(R a){ rep(i,0,len-1) if((*this)[i]==a)return true;  return false; }
    auto begin(){ return view1dIter<T,S,R>(this,0); }
    auto end(){ return view1dIter<T,S,R>(this,len); }
    /*---- view設定 ----*/
    view1d<T,S,R> &dmy(R dmy_) { dummy=dmy_; return *this; }//ダミー値セット
    template<class...Q> view1d<T,S,R> &st(Q... s_){//始点set 負は末尾からの位置
        this->s=RevIfNeg(SllD(s_...));
        this->len=AutoLen(this->s,this->d,this->dsize);
        return *this;
    }
    template<class...Q> view1d<T,S,R> &en(Q... e_){//終了条件再設定 負は末尾から
        this->len=Len(s, RevIfNeg(SllD(e_...)), d);
        return *this;
    }
    template<class...Q> view1d<T,S,R> &dir(Q... d_){//方向set、長さはdata端まで
        this->d=SllD(d_...);
        this->len=AutoLen(this->s,this->d,this->dsize);
        return *this;
    }
    template<class...Q>
    view1d<T,S,R> &mv(Q... s_){ this->s+=SllD(s_...); return *this; }//平行移動
    view1d<T,S,R> &size(ll len_) { len=len_; return *this; }//長さset
    template<class Q> view1d<T,S,R> &size(Q &v){ len=(ll)v.size(); return *this; }//長さset
    view1d<T,S,R> &rev() { s+=d*(len-1); d*=-1; return *this; }//反転
    /*---- utility ----*/
    template<class Q> inline static ll sz(Q &v){ return (ll)v.size(); }
    Sll RevIfNeg(Sll pos){//!< 負なら末尾からの位置に変換
        rep(i,0,S-1)  if (pos[i]<0) pos[i]+=dsize[i];
        return pos;
    }
    static ll AutoLen(Sll s_,Sll d_,Sll dsz){//位置s_から方向d_ではみ出すまでの長さ
        Sll e=dsz-1;
        rep(i,0,S-1) e[i] *= (d_[i]>=0);//方向が負の軸を0にする
        return Len(s_,e,d_);
    }
    /*---- 先頭位置s、方向d、終了条件eから長さlen計算 ----*/
    //template<size_t Q> static ll Len(array<ll,Q> s,array<ll,Q> e,array<ll,Q> d){
    //  ll ret=INF;
    //  rep(i,0,Q-1) chmin(ret, Len1(s[i],e[i],d[i]));
    //  return ret;
    //}
    static ll Len(Sll s,Sll e,Sll d){
        ll ret=INF;
        rep(i,0,S-1) chmin(ret, Len1(s[i],e[i],d[i]));
        return ret;
    }
    static ll Len1(ll s,ll e,ll d){
        if (d==0) return INF;
        if (d<0){ s=-s; e=-e; d=-d; }
        if (s>e) return 0;
        return (e-s)/d+1;
    }
    /*---- 可変長引数をSllに変換 ----*/
    template<class...Q> static Sll SllD(Q... args){ return SllRec(0,args...); }
    template<class...Q> static Sll SllRec(ll i,ll first,Q... rest) {
        Sll sll = (i==S-1) ? Sll() : SllRec(i+1,rest...);
        sll[i]=first;
        return sll;
    }
    static Sll SllRec(ll i){ return Sll(); }
    /*---- dataの位置posの値取得 ----*/
    const R &OrgAt(Sll pos) const {
        rep(i,0,S-1){
            if (pos[i]<0 || dsize[i]<=pos[i]) return dummy;
        }
        return OrgAt_(data,pos);
    }
    template<class Q> using V   = vector<Q>;
    template<class Q> using VV  = V<V<Q>>;
    template<class Q> using VVV = V<V<V<Q>>>;
    using Vs  = V<string>;
    using VVs = VV<string>;
    using ll1 = array<ll,1>;
    using ll2 = array<ll,2>;
    using ll3 = array<ll,3>;
    auto &OrgAt_(V<R>   &dat,ll1 pos)const{ auto [i]    =pos; return dat[i]; }
    auto &OrgAt_(string &dat,ll1 pos)const{ auto [i]    =pos; return dat[i]; }
    auto &OrgAt_(VV<R>  &dat,ll2 pos)const{ auto [i,j]  =pos; return dat[i][j]; }
    auto &OrgAt_(Vs     &dat,ll2 pos)const{ auto [i,j]  =pos; return dat[i][j]; }
    auto &OrgAt_(VVV<R> &dat,ll3 pos)const{ auto [i,j,k]=pos; return dat[i][j][k]; }
    auto &OrgAt_(VVs    &dat,ll3 pos)const{ auto [i,j,k]=pos; return dat[i][j][k]; }
    /*---- dataの各軸size取得 ----*/
    static void SetDsize(V<R>   &dat,ll1 &dsz){ dsz={sz(dat)}; }
    static void SetDsize(string &dat,ll1 &dsz){ dsz={sz(dat)}; }
    static void SetDsize(VV<R>  &dat,ll2 &dsz){ dsz={sz(dat),sz(dat[0])}; }
    static void SetDsize(Vs     &dat,ll2 &dsz){ dsz={sz(dat),sz(dat[0])}; }
    static void SetDsize(VVV<R> &dat,ll3 &dsz){ dsz={sz(dat),sz(dat[0]),sz(dat[0][0])}; }
    static void SetDsize(VVs    &dat,ll3 &dsz){ dsz={sz(dat),sz(dat[0]),sz(dat[0][0])}; }
    typedef view1dIter<T,S,R> iterator;
    using value_type = R;
};
template<class Q> using V   = vector<Q>;
template<class Q> using VV  = V<V<Q>>;
template<class Q> using VVV = V<V<V<Q>>>;
template<class T>         view1d(    VVV<T>      ) -> view1d<    VVV<T>,3,T>;
template<class T>         view1d(     VV<T>      ) -> view1d<     VV<T>,2,T>;
template<class T>         view1d(      V<T>      ) -> view1d<      V<T>,1,T>;
;                         view1d(VV<string>      ) -> view1d<VV<string>,3,char>;
;                         view1d( V<string>      ) -> view1d< V<string>,2,char>;
;                         view1d(   string       ) -> view1d<   string ,1,char>;
template<class T,class S> view1d(    VVV<T>,S,S,S) -> view1d<    VVV<T>,3,T>;
template<class T,class S> view1d(     VV<T>,S,S,S) -> view1d<     VV<T>,2,T>;
template<class T,class S> view1d(      V<T>,S,S,S) -> view1d<      V<T>,1,T>;
template<class S>         view1d(VV<string>,S,S,S) -> view1d<VV<string>,3,char>;
template<class S>         view1d( V<string>,S,S,S) -> view1d< V<string>,2,char>;
template<class S>         view1d(   string ,S,S,S) -> view1d<   string ,1,char>;
template<class VIEW2D> struct view2dIter{
    VIEW2D *vw=nullptr;
    ll idx=INF;
    view2dIter(){}
    view2dIter(VIEW2D *vw_,ll idx_):vw(vw_),idx(idx_){}
    view2dIter(const view2dIter &it):vw(it.vw),idx(it.idx){}
    auto &operator*()          { return (*vw)[x(idx)][y(idx)]; }
    auto &operator*()     const{ return (*vw)[x(idx)][y(idx)]; }
    auto &operator[](ll i)     { return (*vw)[x(idx+i)][y(idx+i)]; }
    auto &operator[](ll i)const{ return (*vw)[x(idx+i)][y(idx+i)]; }
    auto &operator++(){ idx++; return *this; }
    auto &operator--(){ idx--; return *this; }
    auto operator++(int){ auto it=*this; idx++; return it; }
    auto operator--(int){ auto it=*this; idx--; return it; }
    auto &operator+=(ll n){ idx+=n; return *this; }
    auto &operator-=(ll n){ idx-=n; return *this; }
    auto operator+(ll n){ auto it=*this; return it+=n; }
    auto operator-(ll n){ auto it=*this; return it-=n; }
    ll   operator- (const view2dIter &it)const{ return idx- it.idx; }
    bool operator< (const view2dIter &it)const{ return idx< it.idx; }
    bool operator> (const view2dIter &it)const{ return idx> it.idx; }
    bool operator<=(const view2dIter &it)const{ return idx<=it.idx; }
    bool operator>=(const view2dIter &it)const{ return idx>=it.idx; }
    bool operator!=(const view2dIter &it)const{ return idx!=it.idx; }
    bool operator==(const view2dIter &it)const{ return idx==it.idx; }
    ll x(ll i)const{ return i/vw->leny(); }
    ll y(ll i)const{ return i%vw->leny(); }
    using iterator_category = random_access_iterator_tag;
    using value_type = typename VIEW2D::value_type;
    using difference_type = ll;
    using pointer = value_type*;
    using reference = value_type&;
};
template<class T,size_t S,class R> struct view2d{
    using Sll=array<ll,S>;
    T &data; //参照先データ
    Sll dsize; //参照先データ各軸のサイズ
    Sll s=Sll(); //view始点
    Sll dx=Sll(); //view x軸方向
    Sll dy=Sll(); //view y軸方向
    ll xl; //view x軸長さ
    ll yl; //view y軸長さ
    R dummy=R(); //範囲外の値
    /*---- コンストラクタ ----*/
    view2d(T &v): data(v){
        view1d<T,S,R>::SetDsize(v,dsize);
        dx[S-2]=1;
        dy[S-1]=1;
        xl=dsize[S-2];
        yl=dsize[S-1];
    }
    /*---- 演算 ----*/
    //template<class Q> auto &operator =(const Q &b){ rep(i,0,xl-1) (*this)[i] =b; return *this; }
    //template<class Q,size_t P,class O> auto &operator+=(const view2d<Q,P,O> &b){
    //  rep(i,0,xl-1) (*this)[i]
    //  return PlSeq(b); }
    /*---- getter ----*/
    ll size() const { return xl; }
    array<ll,2> shape() const { return {xl,yl}; }
    ll lenx() const { return xl; }
    ll leny() const { return yl; }
    ///pll shape() const { return {xl,yl}; }
    view1d<T,S,R> operator[](ll i){ return view1d(data,s+dx*i,dy,yl,dummy); }
    const view1d<T,S,R> operator[](ll i) const { return view1d(data,s+dx*i,dy,yl,dummy); }
    R &at(ll i,ll j){ if(i<0)i+=xl;  if(j<0)j+=yl;  return (*this)[i][j]; }
    //vector<vector<R>> tovv(){
    //  vector<vector<R>> vvv(xl);
    //  ///rep(i,0,xl-1) vvv[i]=(*this)[i].tov();  return vvv;
    //  rep(i,0,xl-1) vvv[i]=(*this)[i];  return vvv;
    //}
    operator vector<vector<R>>(){
        vector<vector<R>> v(xl);
        rep(i,0,xl-1){ v[i]=(*this)[i]; }  return v;
    }
    auto begin(){ return view2dIter(this,0); }
    auto end(){ return view2dIter(this,xl*yl); }
#if defined(_DEBUG)
    void dump(){ ::dump(tovv()); }
#endif
    /*---- view設定 ----*/
    view2d<T,S,R> &dmy(R dmy_) { dummy=dmy_; return *this; }//ダミー値セット
    template<class...Q> view2d<T,S,R> &st(Q... s_){//始点set 負は末尾からの位置
        this->s=RevIfNeg(view1d<T,S,R>::SllD(s_...));
        this->xl=view1d<T,S,R>::AutoLen(this->s,this->dx,this->dsize);
        this->yl=view1d<T,S,R>::AutoLen(this->s,this->dy,this->dsize);
        return *this;
    }
    template<class...Q> view2d<T,S,R> &dirx(Q... d_){//x軸set、長さはdata端まで
        this->dx=view1d<T,S,R>::SllD(d_...);
        this->xl=view1d<T,S,R>::AutoLen(this->s,this->dx,this->dsize);
        return *this;
    }
    template<class...Q> view2d<T,S,R> &diry(Q... d_){//y軸set、長さはdata端まで
        this->dy=view1d<T,S,R>::SllD(d_...);
        this->yl=view1d<T,S,R>::AutoLen(this->s,this->dy,this->dsize);
        return *this;
    }
    template<class...Q> view2d<T,S,R> &mv(Q... s_){//平行移動
        this->s += view1d<T,S,R>::SllD(s_...);
        return *this;
    }
    view2d<T,S,R> &lenx(ll xl_) { xl=xl_; return *this; }
    view2d<T,S,R> &leny(ll yl_) { yl=yl_; return *this; }
    view2d<T,S,R> &shape(ll xl_,ll yl_) { xl=xl_; yl=yl_; return *this; }
    template<class Q> view2d<T,S,R> &shape(Q &v) { xl=v.lenx(); yl=v.leny(); return *this; }
    view2d<T,S,R> &rot90 () { s+=dx*(xl-1); swap(xl,yl); swap(dx,dy); dy*=-1; return *this; }
    view2d<T,S,R> &rot270() { s+=dy*(yl-1); swap(xl,yl); swap(dx,dy); dx*=-1; return *this; }
    view2d<T,S,R> &rot180() { s+=dx*(xl-1)+dy*(yl-1); dx*=-1; dy*=-1; return *this; }
    view2d<T,S,R> &revx() { s+=dx*(xl-1); dx*=-1; return *this; }
    view2d<T,S,R> &revy() { s+=dy*(yl-1); dy*=-1; return *this; }
    view2d<T,S,R> &swapxy() { swap(xl,yl); swap(dx,dy); return *this; }//xy軸入替
    /*---- utility ----*/
    Sll RevIfNeg(Sll pos){//!< 負なら末尾からの位置に変換
        rep(i,0,S-1)  if (pos[i]<0) pos[i]+=dsize[i];
        return pos;
    }
    using value_type = R;
};
template<class T> view2d(    VVV<T>) -> view2d<    VVV<T>,3,T>;
template<class T> view2d(     VV<T>) -> view2d<     VV<T>,2,T>;
;                 view2d(VV<string>) -> view2d<VV<string>,3,char>;
;                 view2d( V<string>) -> view2d< V<string>,2,char>;
/*
zipは参照のpairを返す。それをさらに参照するのはまずいので、コピーする。
*/
template<class ZIP> struct zipIter{
    ZIP *z=nullptr;
    ll idx=INF;
    zipIter(){}
    zipIter(ZIP *z_,ll idx_):z(z_),idx(idx_){}
    zipIter(const zipIter<ZIP> &it):z(it.z),idx(it.idx){}
    auto operator*()          { return (*z)[idx]; }//参照ではなく値戻し
    auto operator*()     const{ return (*z)[idx]; }//同上
    auto operator[](ll i)     { return (*z)[idx+i]; }//同上
    auto operator[](ll i)const{ return (*z)[idx+i]; }//同上
    auto &operator++(){ idx++; return *this; }
    auto &operator--(){ idx--; return *this; }
    auto operator++(int){ auto it=*this; idx++; return it; }
    auto operator--(int){ auto it=*this; idx--; return it; }
    auto &operator+=(ll n){ idx+=n; return *this; }
    auto &operator-=(ll n){ idx-=n; return *this; }
    auto operator+(ll n){ auto it=*this; return it+=n; }
    auto operator-(ll n){ auto it=*this; return it-=n; }
    ll   operator- (const zipIter<ZIP> &it)const{ return idx- it.idx; }
    bool operator< (const zipIter<ZIP> &it)const{ return idx< it.idx; }
    bool operator> (const zipIter<ZIP> &it)const{ return idx> it.idx; }
    bool operator<=(const zipIter<ZIP> &it)const{ return idx<=it.idx; }
    bool operator>=(const zipIter<ZIP> &it)const{ return idx>=it.idx; }
    bool operator!=(const zipIter<ZIP> &it)const{ return idx!=it.idx; }
    bool operator==(const zipIter<ZIP> &it)const{ return idx==it.idx; }
    using iterator_category = random_access_iterator_tag;
    using value_type = typename ZIP::value_type;
    using difference_type = ll;
    using pointer = value_type*;
    using reference = value_type&;
};
template<class T,class S> struct zip2{
    T &t; S &s;
    zip2(T &t_,S &s_):t(t_),s(s_){}
    ll size() const { return t.end()-t.begin(); }
    auto operator[](ll i)     { return make_pair(ref(t.begin()[i]),ref(s.begin()[i])); }
    auto operator[](ll i)const{ return make_pair(ref(t.begin()[i]),ref(s.begin()[i])); }
    auto begin(){ return zipIter(this,0); }
    auto end(){ return zipIter(this,size()); }
    using value_type = pair<typename T::value_type,typename S::value_type>;
};
template<class T,class S,class R> struct zip3{
    T &t; S &s; R &r;
    zip3(T &t_,S &s_,R &r_):t(t_),s(s_),r(r_){}
    ll size() const { return t.end()-t.begin(); }
    auto operator[](ll i)     { return make_tuple(ref(t.begin()[i]),ref(s.begin()[i]),ref(r.begin()[i])); }
    auto operator[](ll i)const{ return make_tuple(ref(t.begin()[i]),ref(s.begin()[i]),ref(r.begin()[i])); }
    auto begin(){ return zipIter(this,0); }
    auto end(){ return zipIter(this,size()); }
    using value_type = tuple<typename T::value_type,typename S::value_type,typename R::value_type>;
};
#endif//テンプレートend
#if defined(_MSC_VER) && __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
#endif
#if defined(_MSC_VER)
#pragma warning( disable : 4996 )
#endif
struct{
    system_clock::time_point st = system_clock::now();
    ll operator()()const{return duration_cast<microseconds>(system_clock::now()-st).count()/1000;}
} timeget;
//////////////////////////////////////////
struct cinutil{
    template<class T> static void cin1core(T &a){ cin>>a; }
    template<class T,class S> static void cin1core(pair<T,S> &a){
        cin1core(a.first), cin1core(a.second);
    }
    template<class... Args> static void cin1core(tuple<Args...> &a){
        cinTplRec<tuple<Args...>,sizeof...(Args)-1>()(a);
    }
private:
    template<class Tpl,int i> struct cinTplRec{
        void operator()(Tpl &a){ cinTplRec<Tpl,i-1>()(a); cin1core(get<i>(a)); }
    };
    template<class Tpl> struct cinTplRec<Tpl,0>{
        void operator()(Tpl &a){ cin1core(get<0>(a)); }
    };
};
template<class T> T cin1(){ T a; cinutil::cin1core(a); return a; }
template<class... Args> tuple<Args...> cins(){ return cin1<tuple<Args...>>(); }
struct fraction{
    ll bunsi=0,bunbo=1;
    /*---- utilty ----*/
    fraction &norm(){//正規化(約分,分母正)
        ll g=gcd(bunsi,bunbo);
        bunsi/=g,bunbo/=g;
        if (bunbo<0) bunsi*=-1,bunbo*=-1;
        return *this;
    }
    fraction &invsg(){//正負反転
        bunsi*=-1;
        return *this;
    }
    /*---- I/F ----*/
    fraction(){}
    fraction(ll bunsi_,ll bunbo_):bunsi(bunsi_),bunbo(bunbo_){
        norm();
    }
    fraction(ll b):bunsi(b){}//整数用コンストラクタ
    fraction(ll bunsi_,ll bunbo_,bool)//正規化無のコンストラクタ
        :bunsi(bunsi_),bunbo(bunbo_){}
    fraction &operator+=(fraction b){
        ll bo=bunbo*b.bunbo;
        ll si=bunsi*b.bunbo + b.bunsi*bunbo;
        bunsi=si,bunbo=bo;
        return norm();
    }
    fraction &operator-=(fraction b){
        return (*this)+=b.invsg();
    }
    fraction &operator*=(fraction b){
        bunsi*=b.bunsi;
        bunbo*=b.bunbo;
        return norm();
    }
    fraction &operator/=(fraction b){
        return (*this)*=b.inv();
    }
    fraction &operator+=(ll b){ bunsi+=bunbo*b; return *this; }
    fraction &operator-=(ll b){ return (*this)+=-b; }
    fraction &operator*=(ll b){ bunsi*=b; return norm(); }
    fraction &operator/=(ll b){ bunbo*=b; return norm(); }
    fraction operator-()const{ return {-bunsi,bunbo,0}; }
    fraction operator+(const fraction &b)const{ return fraction(*this)+=b; }
    fraction operator-(const fraction &b)const{ return fraction(*this)-=b; }
    fraction operator*(const fraction &b)const{ return fraction(*this)*=b; }
    fraction operator/(const fraction &b)const{ return fraction(*this)/=b; }
    fraction operator+(ll b)const{ return fraction(*this)+=b; }
    fraction operator-(ll b)const{ return fraction(*this)-=b; }
    fraction operator*(ll b)const{ return fraction(*this)*=b; }
    fraction operator/(ll b)const{ return fraction(*this)/=b; }
    friend fraction operator+(ll a,const fraction &b){ return b+a; }
    friend fraction operator-(ll a,const fraction &b){ return -b+a; }
    friend fraction operator*(ll a,const fraction &b){ return b*a; }
    friend fraction operator/(ll a,const fraction &b){ return b.inv()*a; }
    bool operator==(const fraction &b)const{
        return bunsi==b.bunsi && bunbo==b.bunbo;
    }
    bool operator!=(const fraction &b)const{ return !(*this==b); }
    bool operator<(const fraction &b)const{
        return (b-*this).bunsi>0;
    }
    bool operator>(const fraction &b)const{ return b<*this; }
    bool operator>=(const fraction &b)const{ return !(*this<b); }
    bool operator<=(const fraction &b)const{ return b>=*this; }
    friend ostream &operator << (ostream &os,const fraction &a){
        return os<<a.bunsi<<'/'<<a.bunbo;
    }
    fraction inv()const{
        if (bunsi<0) return {-bunbo,-bunsi,0};
        else         return {bunbo, bunsi,0};
    }
};
/*
- 定義   ↓b分のa (a/b)
fraction x(a,b);
.        ↓整数a
fraction x(a);
.        ↓正規化無(約分と分母正化なしで良い時)
fraction x(a,b,true);
fraction y=x.inv(); 逆数
x+y,x-y,x*y,x/y,-x,  x+=y,x-=y,x*=y,x/=y,  x==y,x!=y,x<y,x>y,x<=y,x>=y
cout << x; //"a/b"と表示
*/
template<long long MOD> struct mll_{
    using Int = long long;
    using ll = long long;
    ll val_=0;
    /*---- utility ----*/
    mll_ &norm(){ return normR().normS(); }//正規化
    mll_ &normR(){ val_%=MOD; return *this; }//剰余正規化のみ
    mll_ &normS(){ if (val_<0) val_+=MOD; return *this; }//正負正規化のみ
    mll_ &normP(){ if (val_>=MOD) val_-=MOD; return *this; }//加算時正規化
    mll_ &invsg(){ val_=-val_; return normS(); }//正負反転
    ll modinv(int a){//a^-1 mod MOD
        int ypre=0,y=1,apre=MOD;
        while (a>1){
            int t=apre/a;
            apre-=a*t,swap(a,apre);
            ypre-=y*t,swap(y,ypre);
        }
        return y<0 ? y+MOD: y;
    }
    /*---- I/F ----*/
    mll_(){}
    mll_(ll v): val_(v){ norm(); }
    mll_(ll v,bool b): val_(v){} //正規化無のコンストラクタ
    Int val()const{ return (Int)val_; }
    bool isnone() const { return val_==-1; } //true:値なし
    mll_ &none() { val_=-1; return *this; } //値なしにする
    mll_ &inv(){ val_=modinv((int)val_); return *this; }
    mll_ &operator+=(mll_ b){ val_+=b.val_; return normP(); }
    mll_ &operator-=(mll_ b){ val_-=b.val_; return normS(); }
    mll_ &operator*=(mll_ b){ val_*=b.val_; return normR(); }
    mll_ &operator/=(mll_ b){ return *this*=b.inv(); }
    mll_ &operator+=(ll b){ return *this+=mll_(b); }
    mll_ &operator-=(ll b){ return *this-=mll_(b); }
    mll_ &operator*=(ll b){ return *this*=mll_(b); }
    mll_ &operator/=(ll b){ return *this/=mll_(b); }
    mll_ operator-()const{ return mll_(*this).invsg(); }
    mll_ operator+(mll_ b)const{ return mll_(*this)+=b; }
    mll_ operator-(mll_ b)const{ return mll_(*this)-=b; }
    mll_ operator*(mll_ b)const{ return mll_(*this)*=b; }
    mll_ operator/(mll_ b)const{ return mll_(*this)/=b; }
    mll_ operator+(ll b)const{ return mll_(*this)+=b; }
    mll_ operator-(ll b)const{ return mll_(*this)-=b; }
    mll_ operator*(ll b)const{ return mll_(*this)*=b; }
    mll_ operator/(ll b)const{ return mll_(*this)/=b; }
    friend mll_ operator+(ll a,mll_ b){ return b+a; }
    friend mll_ operator-(ll a,mll_ b){ return -b+a; }
    friend mll_ operator*(ll a,mll_ b){ return b*a; }
    friend mll_ operator/(ll a,mll_ b){ return mll_(a)/b; }
    bool operator==(mll_ b)const{ return val_==b.val_; }
    bool operator!=(mll_ b)const{ return val_!=b.val_; }
    bool operator==(ll b)const{ return *this==mll_(b); }
    bool operator!=(ll b)const{ return *this!=mll_(b); }
    friend bool operator==(ll a,mll_ b){ return mll_(a)==b; }
    friend bool operator!=(ll a,mll_ b){ return mll_(a)!=b; }
    friend ostream &operator<<(ostream &os,mll_  a){ return os << a.val_; }
    friend istream &operator>>(istream &is,mll_ &a){ return is >> a.val_; }
    mll_ pow(ll k)const{
        mll_ ret(1,false),a(*this);
        for (; k>0; k>>=1,a*=a) if (k&1)ret*=a;
        return ret;
    }
    static constexpr int mod() { return MOD; }
    //enum{ modll=MOD };
};
#if 0
#define MODLL (1000000007LL)
#else
#define MODLL (998244353LL)
#endif
using mll = mll_<MODLL>;
//using mll = fraction;
using vmll    = std::vector<mll>;
using vvmll   = std::vector<vmll>;
using vvvmll  = std::vector<vvmll>;
using vvvvmll = std::vector<vvvmll>;
template<class T> struct Vector: vector<T>{
    using Int = long long;
    using vT=vector<T>;
    using cvT=const vector<T>;
    using cT=const T;
    using vT::vT; //親クラスのコンストラクタの隠蔽を回避
    using vT::begin,vT::end,vT::insert,vT::erase;
    auto it(Int i){ return begin()+i; }
    auto it(Int i)const{ return begin()+i; }
    Vector(cvT& b):vT(b){}
    Vector(vT&& b):vT(move(b)){}
    template<class S> Vector(const Vector<S>& b):vT(b.begin(),b.end()){}
    template<class S> Vector(const vector<S>& b):vT(b.begin(),b.end()){}
    Vector(Int n,T s,T d){ iota(n,s,d); }
    Vector(Int n,function<T(Int)> g):vT(n){ for (Int i=0;i<n;++i) (*this)[i]=g(i); }
    template<class S> Vector &operator+=(S x){ for (T &e: *this) e+=x;  return *this; }
    template<class S> Vector &operator-=(S x){ for (T &e: *this) e-=x;  return *this; }
    template<class S> Vector &operator*=(S x){ for (T &e: *this) e*=x;  return *this; }
    template<class S> Vector &operator/=(S x){ for (T &e: *this) e/=x;  return *this; }
    template<class S> Vector &operator%=(S x){ for (T &e: *this) e%=x;  return *this; }
    template<class S> Vector operator+(S x)const{ return Vector(*this)+=x; }
    template<class S> Vector operator-(S x)const{ return Vector(*this)-=x; }
    template<class S> Vector operator*(S x)const{ return Vector(*this)*=x; }
    template<class S> Vector operator/(S x)const{ return Vector(*this)/=x; }
    template<class S> Vector operator%(S x)const{ return Vector(*this)%=x; }
    Vector operator-()const{ return Vector(*this)*=-1; }
    template<class S> friend Vector operator-(S x,const Vector &a){ return -a+=x; }
    Int size()const{ return (Int)vT::size(); }
    Vector &push_back(cT& x){ vT::push_back(x); return *this; }
    Vector &pop_back(){ vT::pop_back(); return *this; }
    Vector &push_front(cT& x){ insert(begin(),x); return *this; }
    Vector &pop_front(){ erase(0); return *this; }
    Vector &insert(Int i,cT& x){ insert(it(i),x); return *this; }
    Vector &insert(Int i,cvT& b){ insert(it(i),b.begin(),b.end()); return *this; }
    Vector &erase(Int i){ erase(it(i)); return *this; }
    Vector &erase(Int l,Int r){ erase(it(l),it(r+1)); return *this; }
    Vector &concat(cvT &b){ insert(end(),b.begin(),b.end()); return *this; }
    Vector &concat(cvT &b,Int n){ for (int i=0;i<n;++i) concat(b); return *this; }
    Vector &repeat(Int n){ concat(vT(*this),n-1); return *this; }
    Vector &reverse(){ std::reverse(begin(),end()); return *this; }
    Vector &sort(){ std::sort(begin(),end()); return *this; }
    Vector &rsort(){ std::sort(vT::rbegin(),vT::rend()); return *this; }
    template<class Pr> Vector &sort(Pr pr){ std::sort(begin(),end(),pr); return *this; }
    Vector &uniq(){ erase(unique(begin(),end()),end()); return *this; }
    Vector &sortq(){ return sort().uniq(); }
    Vector &fill(Int l,Int r,cT& x){ std::fill(it(l),it(r+1),x); return *this; }
    template<class S=Int> Vector &iota(Int n,T s=0,S d=1){
        vT::resize(n);
        if (n==0) return *this;
        (*this)[0]=s;
        for (int i=1;i<n;++i) (*this)[i]=(*this)[i-1]+d;
        return *this;
    }
    Int count(cT& x)const{ return Int(std::count(begin(),end(),x)); }
    Int count(Int l,Int r,cT& x)const{ return Int(std::count(it(l),it(r+1),x)); }
    template<class Pr> Int countif(Pr pr)const{ return Int(count_if(begin(),end(),pr)); }
    template<class Pr> Int countif(Int l,Int r,Pr pr)const{ return Int(count_if(it(l),it(r+1),pr)); }
    Int find(cT& x)const{ return Int(std::find(begin(),end(),x)-begin()); }
    Int find(Int l,Int r,cT& x)const{ return Int(std::find(it(l),it(r+1),x)-begin()); }
    template<class Pr> Int findif(Pr pr)const{ return Int(find_if(begin(),end(),pr)-begin()); }
    template<class Pr> Int findif(Int l,Int r,Pr pr)const{ return Int(find_if(it(l),it(r+1),pr)-begin()); }
    Vector<Int> findall(cT& x)const{ return findall(0,size()-1,x); }
    Vector<Int> findall(Int l,Int r,cT& x)const{ return findallif(l,r,[&](cT& y){return y==x;}); }
    template<class Pr> Vector<Int> findallif(Pr pr)const{ return findallif(0,size()-1,pr); }
    template<class Pr> Vector<Int> findallif(Int l,Int r,Pr pr)const{
        Vector<Int> ret;
        for (Int i=l;i<=r;++i) if (pr((*this)[i])) ret.push_back(i);
        return ret;
    }
    Int  flooridx(cT& x)const{ return Int(upper_bound(begin(),end(),x)-begin()-1); }
    Int   ceilidx(cT& x)const{ return Int(lower_bound(begin(),end(),x)-begin()); }
    Int  leftnmof(cT& x)const{ return flooridx(x)+1; }
    Int rightnmof(cT& x)const{ return size()-ceilidx(x); }
    bool contains(cT& x)const{ Int i=flooridx(x); return i>=0 && (*this)[i]==x; }
    template<class Pr> Int  flooridx(cT& x,Pr pr)const{ return Int(upper_bound(begin(),end(),x,pr)-begin()-1); }
    template<class Pr> Int   ceilidx(cT& x,Pr pr)const{ return Int(lower_bound(begin(),end(),x,pr)-begin()); }
    template<class Pr> Int  leftnmof(cT& x,Pr pr)const{ return flooridx(x,pr)+1; }
    template<class Pr> Int rightnmof(cT& x,Pr pr)const{ return size()-ceilidx(x,pr); }
    template<class Pr> bool contains(cT& x,Pr pr)const{ Int i=flooridx(x,pr); return i>=0 && (*this)[i]==x; }
};
/*
vll a={9,8,7},b={1,2,3};
vpll p={{5,3},{7,8},{0,2},};
- -------- 操作系 --------
a+=x a-=x a*=x a/=x a%=x a+x a-x a*x a/x a%x -a x-a  //∀i a[i]にxを演算
a.push_front(x);
a.push_back(x);
a.pop_front();
a.pop_back();
a.insert(i,x); //a[i]にx挿入
a.insert(i,b); //a[i]にvll b挿入
a.erase(i);    //a[i]削除
a.erase(l,r);  //区間[l,r]削除
a.concat(b);   //aにbを結合
a.concat(b,n); //aにbをn回結合
a.repeat(n);   //aをn回繰り返す
a.reverse(); //反転
a.sort();    //ソート
a.rsort();   //逆順ソート
p.sort([&](pll x,pll y){return x.second<y.second;});//比較関数指定ソート
a.uniq();      //連続同値を1つにする
a.sortq();     //ソートしてユニーク
a.fill(l,r,x); //[l,r]にx代入
a.iota(n,s,d); //aを等差数列にする 長さn,初項s,公差d
vll a(n,s,d);  //コンストラクタ版iota
- -------- 検索系 --------
auto pr=[&](auto &x){ return x>0; };  //検索条件
ll m=a.count(x);     //xの個数
ll m=a.count(l,r,x); //xの個数in[l,r]
ll m=a.countif(pr);     //条件満たす個数
ll m=a.countif(l,r,pr); //条件満たす個数in[l,r]
ll i=a.find(x);         //xの最左位置i         ない時N(配列長)
ll i=a.find(l,r,x);     //xの最左位置i in[l,r] ない時r+1
ll i=a.findif(pr);      //条件満たす最左位置i         ない時N(配列長)
ll i=a.findif(l,r,pr);  //条件満たす最左位置i in[l,r] ない時r+1
vll is=a.findall(x);        //xの位置i列挙
vll is=a.findall(l,r,x);    //xの位置i列挙in[l,r]
vll is=a.findallif(pr);     //条件満たす位置i列挙
vll is=a.findallif(l,r,pr); //条件満たす位置i列挙in[l,r]
- -------- 昇順sort済み配列用 --------
ll i=a.flooridx(x);   //x以下の最近傍位置i ない時-1
ll i=a.ceilidx(x);    //x以上の最近傍位置i ない時N(配列長)
ll m=a.leftnmof(x);   //x以下の個数
ll m=a.rightnmof(x);  //x以上の個数
bool b=a.contains(x); //xを含む
- -------- 比較関数prでsort済みの配列用 --------
auto pr=[&](auto &x,auto &y){ return x>y; }; //降順ソート時
ll i=a.flooridx(x,pr);   //x以左の最近傍位置i ない時-1
ll i=a.ceilidx(x,pr);    //x以右の最近傍位置i ない時N(配列長)
ll m=a.leftnmof(x,pr);   //x以左の個数
ll m=a.rightnmof(x,pr);  //x以右の個数
bool b=a.contains(x,pr); //xを含む
a.concat(b,n).pop_back().rsort().uniq();  //連続適用できる
*/
namespace SolvingSpace{
template<class T> using vector = Vector<T>;
using    vll=vector<   ll>; using    vmll=vector<   mll>; using   vdd=vector<  dd>;
using   vvll=vector<  vll>; using   vvmll=vector<  vmll>; using  vvdd=vector< vdd>;
using  vvvll=vector< vvll>; using  vvvmll=vector< vvmll>; using vvvdd=vector<vvdd>;
using vvvvll=vector<vvvll>; using vvvvmll=vector<vvvmll>;
using   vpll=vector<  pll>; using    vtll=vector<   tll>; using  vqll=vector< qll>;
using  vvpll=vector< vpll>; using   vvtll=vector<  vtll>; using vvqll=vector<vqll>;
using vss=vector<string>;
template<class T> vector<T> cinv(ll nm){ return vector<T>(nm,[](ll i){ (void)i; return cin1<T>(); }); }
template<class T> vector<vector<T>> cinvv(ll H,ll W){ return vector<vector<T>>(H,[&](ll i){ (void)i; return cinv<T>(W); }); }
template<class T> struct wrapVector1d{
    using S=typename T::value_type;
    using Int = long long;
    const T *v;
    S Ini;
    wrapVector1d(const T &v_,S ini_=S()):v(&v_),Ini(ini_){}
    S operator[](Int i)const{ return (i<0 || (Int)v->size()<=i) ? Ini : (*v)[i]; }
};
template<class T> struct wrapVector2d{
    using S=typename T::value_type;
    using Int = long long;
    const vector<T> *v;
    S Ini;
    T dmy;
    wrapVector2d(const vector<T> &v_,S ini_=S()):v(&v_),Ini(ini_){}
    wrapVector1d<T> operator[](ll i)const{
        return (i<0 || (Int)v->size()<=i) ?
            wrapVector1d(dmy,Ini) : wrapVector1d((*v)[i],Ini);
    }
};
template<int DIM> struct rangeNd{
    using Int = long long;
    using NInt = array<Int,DIM>;
    NInt sv,ev,lv,dv; //各変数の初期値,終了値,項数,公差(通常±1)
    Int size_=1;
    NInt cyclemask;
    rangeNd(const NInt &sv_,const NInt &ev_,Int cycle=-1): sv(sv_),ev(ev_){
        for (ll i=0;i<DIM;++i) lv[i]=abs(sv[i]-ev[i])+1;
        for (ll i=0;i<DIM;++i) dv[i]=sign(ev[i]-sv[i]);
        for (ll i=0;i<DIM;++i) size_*=lv[i];
        for (ll i=0;i<DIM;++i) cyclemask[i]=~0;
        if (cycle!=-1) cyclemask[0]=cycle-1;
    }
    Int size()const{ return size_; }
    Int csize()const{ return size_/lv[0]; }
    bool isCycle()const{ return cyclemask[0]!=~0; }
    Int cycle()const{ return isCycle() ? cyclemask[0]+1 : -1; }
    tuple<Int,Int,Int,Int> SELDx()const{ return SELDof(0); }
    tuple<Int,Int,Int,Int> SELDy()const{ return SELDof(1); }
    tuple<Int,Int,Int,Int> SELDz()const{ return SELDof(2); }
    tuple<Int,Int,Int,Int> SELDw()const{ return SELDof(3); }
    array<NInt,2> rngs()const{ return {sv,dv}; }
    NInt ls()const{ return lv; }
    NInt v2i(const NInt &vx)const{
        NInt ret;
        for (Int i=0;i<DIM;++i) ret[i]=(vx[i]-sv[i])*dv[i];
        return ret;
    }
    auto toID(){
        return [this](const NInt &vx)->Int{
            Int id=0;
            for (Int i=0;i<DIM;++i){
                Int x=vx[i],s=sv[i],l=lv[i],d=dv[i],xx=(x-s)*d;
                if (xx<0 || l<=xx) return -1;
                id=id*l+(xx&cyclemask[i]);
            }
            return id;
        };
    }
    auto fromID(){
        return [this](Int id)->NInt{
            NInt ret;
            for (Int i=DIM-1;i>=0;--i){
                Int s=sv[i],l=lv[i],d=dv[i],nid=id/l,xx=id-nid*l;
                ret[i]=s+xx*d;
                id=nid;
            }
            return ret;
        };
    }
    bool out(const NInt &vx){
        for (Int i=0;i<DIM;++i){
            Int x=vx[i],s=sv[i],l=lv[i],d=dv[i];
            Int xx=(x-s)*d;
            if (xx<0 || l<=xx) return true;
        }
        return false;
    }
    NInt start(Int xx=0)const{
        NInt ret=sv;
        ret[0]=sv[0]+dv[0]*xx;
        return ret;
    }
    bool next(NInt &vx,Int dMn=0)const{ //次へ進める 対象:第dMn以降 @return true:次有
        for (int d=DIM-1;d>=dMn;--d){
            if (vx[d]==ev[d]) vx[d]=sv[d];
            else{ vx[d]+=dv[d]; return true; }
        }
        return false;
    }
private:
    tuple<Int,Int,Int,Int> SELDof(Int i)const{
        if (i>=DIM) return {-1,-1,-1,-1};
        return {sv[i],ev[i],lv[i],dv[i]};
    }
    static Int sign(Int x){ return x==0 ? 0 : x>0 ? 1 : -1; }
};
template<class STATE,class VAL,class Ope,class Tran,class ToID,class FromID>
struct memoDP{
    using Int = long long;
    using ll = long long;
    Ope ope; //貰うときの演算
    Tran tran;
    ToID toID;
    FromID fromID;
    VAL yet,ini,dmy; //yet:未計算印の値 ini:ope演算の初期値
    vector<VAL> dp_;
    bool autoResize = true;
    bool nextdp = false; //true:nextDP
    //-------- コンストラクタ --------
    memoDP(STATE,VAL,Ope ope_,Tran tran_,ToID toID_,FromID fromID_,VAL yet_,VAL ini_)
        : ope(ope_),tran(tran_),toID(toID_),fromID(fromID_),yet(yet_),ini(ini_),dmy(ini_){}
    //事前メモリ確保用コンストラクタ
    template<class RNG> memoDP(STATE,VAL,Ope ope_,Tran tran_,ToID toID_,
        FromID fromID_,VAL yet_,VAL ini_,const RNG &rng)
        : ope(ope_),tran(tran_),toID(toID_),fromID(fromID_),yet(yet_),ini(ini_),dmy(ini_),
        dp_(rng.isCycle() ? rng.csize()*rng.cycle() : rng.size(),yet),
        autoResize(false),nextdp(rng.isCycle()){}
    //-------- アクセス・計算実行 --------
    template<class... Args> VAL &operator()(Args&&... args){ return Dp({args...}); }
    template<class... Args> VAL calcIfAny(Args&&... args){ return calcIfAnySt({args...}); }
    template<class... Args> void calc(Args&&... args){ calcSt({args...}); }
    template<class RNG> void calcall(const RNG &rng){
        if (nextdp) calcallNextDp(rng);
        else        calcallNormal(rng);
    }
    using   vVAL = vector<  VAL>;  using   vvVAL = vector<  vVAL>;
    using vvvVAL = vector<vvVAL>;  using vvvvVAL = vector<vvvVAL>;
    vVAL    tov(array<Int,1> ls){ return tov(ls[0]); }
    vvVAL   tov(array<Int,2> ls){ return tov(ls[0],ls[1]); }
    vvvVAL  tov(array<Int,3> ls){ return tov(ls[0],ls[1],ls[2]); }
    vvvvVAL tov(array<Int,4> ls){ return tov(ls[0],ls[1],ls[2],ls[3]); }
    vVAL tov(Int xl){
        dp_.resize(xl,yet);  return vVAL(dp_);
    }
    vvVAL tov(Int xl,Int yl){
        dp_.resize(xl*yl,yet);
        vvVAL ret(xl,vVAL(yl));
        for (ll i=0;i<xl;++i) for (ll j=0;j<yl;++j) ret[i][j]=dp_[i*yl+j];
        return ret;
    }
    vvvVAL tov(Int xl,Int yl,Int zl){
        dp_.resize(xl*yl*zl,yet);
        vvvVAL ret(xl,vvVAL(yl,vVAL(zl)));
        for (ll i=0;i<xl;++i) for (ll j=0;j<yl;++j) for (ll k=0;k<zl;++k){
            ret[i][j][k]=dp_[i*yl*zl+j*zl+k];
        }
        return ret;
    }
    vvvvVAL tov(Int xl,Int yl,Int zl,Int wl){
        dp_.resize(xl*yl*zl*wl,yet);
        vvvvVAL ret(xl,vvvVAL(yl,vvVAL(zl,vVAL(wl))));
        for (ll i=0;i<xl;++i) for (ll j=0;j<yl;++j) for (ll k=0;k<zl;++k){
            for (ll l=0;l<wl;++l) ret[i][j][k][l]=dp_[i*yl*zl*wl+j*zl*wl+k*wl+l];
        }
        return ret;
    }
private:
    VAL &Dp(const STATE &vx){ //dp_がreallocされると参照が無効になることに注意
        ll id=toID(vx);
        if (autoResize && (ll)dp_.size()<=id) dp_.resize(id+1,yet);
        return id<0 ? (dmy=ini) : dp_[id];
    }
    VAL calcIfAnySt(const STATE &vx){
        VAL &Dpvx=Dp(vx);
        if (Dpvx!=yet) return Dpvx;
        Dpvx=ini;
        auto src_calc2srcVa=[&](const STATE &src,auto calc2srcVa){
            VAL srcVa=calcIfAnySt(src);
            VAL &DpvxL = autoResize ? Dp(vx) : Dpvx; //resizeすると参照が無効になるため
            VAL nVa=ope(DpvxL,calc2srcVa(srcVa));
            if (DpvxL!=nVa) DpvxL=nVa,preRenew(vx,src);
        };
        auto dpWrap=[this](auto &&... args)->VAL{ return calcIfAny(args...); };
        auto setVal=[&](VAL va){
            (autoResize ? Dp(vx) : Dpvx) = va; //resizeすると参照が無効になるため
        };
        tran(vx,src_calc2srcVa,dpWrap,setVal);
        return autoResize ? Dp(vx) : Dpvx; //resizeすると参照が無効になるため
    }
    void calcSt(const STATE &vx,bool force=false){ //force=true:計算済でも再計算
        VAL &Dpvx=Dp(vx);
        if (!force && Dpvx!=yet) return;
        Dpvx=ini;
        auto src_calc2srcVa=[&](const STATE &src,auto calc2srcVa){
            VAL srcVa=Dp(src);
            VAL nVa=ope(Dpvx,calc2srcVa(srcVa));
            if (Dpvx!=nVa) Dpvx=nVa,preRenew(vx,src);
        };
        auto dpWrap=[this](auto &&... args)->VAL{ return (*this)(args...); };
        auto setVal=[&](VAL va){ Dpvx=va; };
        tran(vx,src_calc2srcVa,dpWrap,setVal);
    }
    template<class RNG> void calcallNormal(const RNG &rng){
        STATE vx=rng.start();
        do calcSt(vx); while (rng.next(vx));
    }
    template<class RNG> void calcallNextDp(const RNG &rng){
        assert(rng.isCycle());
        for (ll xx=0;xx<rng.ls()[0];++xx){
            STATE vx=rng.start(xx);
            bool force = rng.cycle()<=xx;
            do calcSt(vx,force); while (rng.next(vx,1));
        }
    }
    //-------- ダンプ --------
public:
    #if 0
    template<int N>
    void dump(rangeNd<N> &rng,vector<string> labels={},const vector<ll> axes={0,1,2,3}){
        auto ls=rng.ls();
        if (nextdp) ls[0]=rng.cycle();
        dumpstring::dumpNd("",tov(ls),args().rngs(rng.rngs()).tr(axes).labels(labels));
    }
    #endif
    //-------- 最適パス --------
    #if 0
private:
    vector<ll> pre;
    ll preini=-1,predmy=-1;
    ll &Pre(const STATE &vx){
        ll id=toID(vx);
        if ((ll)pre.size()<=id) pre.resize(id+1,preini);
        return id<0 ? (predmy=preini) : pre[id];
    }
    void preRenew(const STATE &vx,const STATE &src){ Pre(vx)=toID(src); }
public:
    template<class... Args> vector<STATE> pathget(Args&&... endargs){
        vector<ll> ids{toID({endargs...})};
        while (ids.back()<(ll)pre.size() && pre[ids.back()]!=preini) ids.push_back(pre[ids.back()]);
        reverse(ids.begin(),ids.end());
        vector<STATE> path;
        for (auto&& id: ids) path.push_back(fromID(id));
        return path;
    }
    #else
    void preRenew(const STATE &vx,const STATE &src){ (void)vx,(void)src; }
    #endif
};
template<class T> struct SegmentTree{
    using F = function<T(T,T)>;
    vector<T> dat; //第i要素 dat[i+n-1]
    ll n; //要素数以上の最小の 2のべき乗
    F f;
    T ti;
    SegmentTree(){}
    SegmentTree(vector<T> &v,F f,T ti){ Init(v,f,ti); }
    void Init(vector<T> &v,F f,T ti){
        this->f=f; this->ti=ti;
        for (n = 1; n < (ll)v.size(); n *= 2);
        dat.resize(2*n-1,ti);
        REP(i,(ll)v.size()) { dat[i+n-1] = v[i]; }//初期化
        DEP(i,n-1) { dat[i] = f(dat[2*i+1],dat[2*i+2]); }//構築
    }
    void set(ll i,T x){ //第i要素にxをセット
        i += n-1;  dat[i] = x;
        while (i>0) { i=(i-1)/2; dat[i]=f(dat[2*i+1],dat[2*i+2]); }
    }
    T get(ll a,ll b) { return Get(a,b+1,0,0,n); }
    T operator [] (ll i) { return dat[i+n-1]; }
    void Dump(ll w=5){
        REP(i,n) for (ll k=i+n-1,m=1,p=1; k>=0; p=m,m*=k%2,k=(k==0) ? -1 : (k-1)/2){
            if (m) cerr << ((k<n-1) ? " ┬ " : "") << setw(w) << dat[k];
            else   cerr << ((p&~m) ? " ┘ " : "");
            if (k==0) cerr << '\n';
        } cerr << '\n';
    }
    T Get(ll a,ll b,ll k,ll l,ll r){ //注目ノードk=[l,r)
        if (r<=a || b<=l) return ti; //完全に外
        if (a<=l && r<=b) return dat[k];//完全に含まれる
        return f(Get(a,b,k*2+1,l,(l+r)/2),Get(a,b,k*2+2,(l+r)/2,r));
    }
    template<class C> ll searchl(ll a,ll b,C c){//[a,p](a≦p≦b)がtrueな最左のp
        auto [pos,ignore]=SearchL(a,c,ti,0,0,n);  return min(pos,b+1);
    }
    template<class C> ll searchr(ll a,ll b,C c){//[p,b](a≦p≦b)がtrueな最右のp
        auto [pos,ignore]=SearchR(b+1,c,ti,0,0,n);  return max(pos,a-1);
    }
    template<class C> pair<ll,T> SearchL(ll a,C c,T s,ll k,ll l,ll r){
        if (r<=a) return {r,s};
        T ss=f(s,dat[k]);
        if (a<=l && !c(ss)) return {r,ss};//現セル丸ごとがfalseの時
        if (n-1 <= k) return {l,s};//葉の時
        ll pos,m=(l+r)/2;
        ;           tie(pos,s)=SearchL(a,c,s,2*k+1,l,m);
        if (pos==m) tie(pos,s)=SearchL(a,c,s,2*k+2,m,r);
        return {pos,s};
    }
    template<class C> pair<ll,T> SearchR(ll b,C c,T s,ll k,ll l,ll r){
        if (b<=l) return {l-1,s};
        T ss=f(dat[k],s);
        if (r<=b && !c(ss)) return {l-1,ss};//現セル丸ごとがfalseの時
        if (n-1 <= k) return {l,s};//葉の時
        ll pos,m=(l+r)/2;
        ;             tie(pos,s)=SearchR(b,c,s,2*k+2,m,r);
        if (pos==m-1) tie(pos,s)=SearchR(b,c,s,2*k+1,l,m);
        return {pos,s};
    }
    ll findpeakl(ll l,ll r){
        T m=get(l,r);
        return searchl(l,r,[&](T x){return f(x,m)==x;});
    }
    ll findpeakr(ll l,ll r){
        T m=get(l,r);
        return searchr(l,r,[&](T x){return f(x,m)==x;});
    }
};
template<class T> struct cumulativesum{
    ll n=0;  vector<T> c;
    cumulativesum():c(1){}
    template<class S> cumulativesum(S &&v) : n((ll)v.size()),c(n+1) { Ini(v); }
    template<class S> void init(S &&v){ n=(ll)v.size(); c.resize(n+1); Ini(v); }
    void add(T x) { n++; c.push_back(c.back()+x); }
    T operator()(ll l,ll r){ return c[max(min(n,r+1),0LL)]-c[min(max(0LL,l),n)]; }
    pll flooridx(T x){
        ll i=upper_bound(c.begin(),c.end(),x)-c.begin()-1;
        return {i-1, (i==-1) ? -INF : c[i]};
    }
    pll ceilidx(T x){
        ll i=lower_bound(c.begin(),c.end(),x)-c.begin();
        return {i-1, (i==n+1) ? INF : c[i]};
    }
    T mx(){//区間和max
        T mn=T(),samx=0;
        for (ll i=1;i<=n;++i){
            chmax(samx,c[i]-mn);
            chmin(mn,c[i]);
        }
        return samx;
    }
    template<class S> void Ini(S &&v) { for (ll i=0;i<n;++i) c[i+1]=c[i]+v[i]; }
};
template<class S> cumulativesum(S) -> cumulativesum<typename remove_reference<S>::type::value_type>;
/*
cumulativesum cm(v);
.      ↓空で複数定義、逐次末尾に追加
vector<cumulativesum<ll>> vc(n);
vc[i].add(x);
ll sm=cm(l,r); //v[l]～v[r]の和
ll sm=cm(4,2); //定義外は0埋めと見なす＆逆順は正負反転するだけで落ちない
.     ↓kに近いインデックスi,和s   k負なら(-2,-INF) k<先頭要素なら(-1,0)
auto[i,s]=cm.flooridx(k);
.                    ↑和上限
.     ↓和下限kに近いインデックスiと和s  k負なら(-1,0) 総和<kなら(n,INF)
auto[i,s]=cm.ceilidx(k);
.                   ↑和下限
ll s=cm.mx(); //区間和の最大値 空集合は0とする
*/
void cin2solve()
{
    auto [N,S,H]=cins<ll,ll,ll>();
    auto xyz_=cinv<tuple<ll,ll,ll>>(N);
    wrapVector1d xyz(xyz_,tll{0,0,0});
    
    cumulativesum<ll> cm;
    {
        vll zv;
        for (auto&&[x,y,z]:xyz_) zv.push_back(z);
        cm=cumulativesum(zv);
    }
    /*
    dp(i,s) 区間iのs=0先頭/1末尾 でのスコアmax
    */
    using STATE=array<ll,2>; //i,s
    using VAL=ll;     VAL yet=-INF*7;
    auto operation=[&](VAL a,VAL b){return max(a,b);};  VAL ini=0;
    vll v(N,ini);
    SegmentTree<ll> sgt1(v, [](ll x,ll y){return max(x,y);}, 0);
    SegmentTree<ll> sgt0(v, [](ll x,ll y){return max(x,y);}, 0);
    auto transition=[&](const STATE &vx,auto src_calc2srcVa,auto dp,auto setVal){
        auto[i,s]=vx;
        auto [x,y,z]=xyz[i];
        if (s==0){
            auto pr=[&](auto &d,auto &e){ return get<1>(d) < get<1>(e); };
            ll j=xyz_.flooridx(tll{0,x-H,0},pr);
            VAL va = j>=0 ? sgt1.get(0,j) : 0;
            setVal(va);
            sgt0.set(i,va+cm(i,N-1));
        }
        else{
            auto pr=[&](auto &d,auto &e){ return get<0>(d) < get<0>(e); };
            ll j=xyz_.ceilidx(tll{y-S,0,0},pr);
            VAL va=sgt0.get(j,i);
            va-=cm(i+1,N-1);
            setVal(va);
            sgt1.set(i,va);
        }
    };
    auto [s1,e1]=pll{0,N-1}; //1つ目の変数の範囲
    auto [s2,e2]=pll{0,1};    //2つ目の変数の範囲
    rangeNd<2> rng({s1,s2},{e1,e2});
    memoDP dp(STATE(),VAL(),operation,transition,rng.toID(),rng.fromID(),yet,ini,rng);
    dp.calcall(rng);
    VAL ans=sgt1.get(0,N-1);
    cout << ans << '\n';
    //using STATE=array<ll,1>; //i
    //using VAL=ll;     VAL yet=-INF*7;
    //auto operation=[&](VAL a,VAL b){return max(a,b);};  VAL ini=0;
    //vll v(N,ini);
    //SegmentTree<ll> sgt(v, [](ll x,ll y){return max(x,y);}, 0);
    //auto transition=[&](const STATE &vx,auto src_calc2srcVa,auto dp,auto setVal){
    //  auto[i]=vx;
    //  auto [x,y,z]=xyz[i];
    //  //dp(現在位置)をdp(…)から計算
    //  if (y-x>S) return;
    //  auto pr=[&](auto &x,auto &y){ return get<1>(x) < get<1>(y); };
    //  ll j=xyz_.flooridx(tll{0,x-H,0},pr);   //x以左の最近傍位置i ない時-1
    //  
    //  VAL va = j>=0 ? sgt.get(0,j) : 0;
    //  va+=z;
    //  //結果セット
    //  setVal(va);
    //  sgt.set(i,va);
    //};
    //auto [s1,e1]=pll{0,N-1}; //1つ目の変数の範囲
    //rangeNd<1> rng({s1,},{e1,});
    //memoDP dp(STATE(),VAL(),operation,transition,rng.toID(),rng.fromID(),yet,ini,rng);
    //dp.calcall(rng);
    //VAL ans=sgt.get(0,N-1);
    //cout << ans << '\n';
    return;
}
};//SolvingSpace
//////////////////////////////////////////
int main(){
#if 1
    //SolvingSpace::labo();
    SolvingSpace::cin2solve();
  //SolvingSpace::generand();
#else
    ll t;  cin >> t;
    rep(i,0,t-1){
        SolvingSpace::cin2solve();
        //SolvingSpace::generand();
    } 
#endif
    cerr << timeget() <<"ms"<< '\n';
    return 0;
}
הההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההההה
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0