#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 __has_include(<atcoder/all>)
#include <atcoder/all>
using namespace atcoder;
#endif
using namespace std;
using ll=long long;
using dd=long double;
using pll=pair<ll,ll>;
using tll=tuple<ll,ll,ll>;
using qll=tuple<ll,ll,ll,ll>;
using ll2=array<ll,2>;
using ll3=array<ll,3>;
using ll4=array<ll,4>;
using namespace chrono;
constexpr ll INF = 1201001001001001001;
struct Fast{ Fast(){ cin.tie(0); ios::sync_with_stdio(false); cout<<fixed<<setprecision(numeric_limits<double>::max_digits10); } } fast;
#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 rep1(N)          for (ll dmyi = 0; dmyi < (N); dmyi++)
#define rep2(i, N)       for (ll i = 0; i < (N); i++)
#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,rep2,rep1)(__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 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 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__))
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 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,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
inline ll Floormod(ll a,ll m){ return Floor(a,m)*m; } //負もOK
inline ll Ceilmod(ll a,ll m){ return Ceil(a,m)*m; } //負もOK
inline ll Mod(ll a,ll m){ ll r=a%m; if(r<0)r+=m; return r; } //負もOK
template<class T> inline T Pow(T a,ll n){ T r=1; for(; n>0; n>>=1,a*=a){ if(n&1)r*=a; } return r; }
inline ll Pow(int a,ll n){ return Pow((ll)a,n); }
inline ll limitmul(ll a,ll b,ll u){ return b==0||a<=u/b ? a*b : u; }//min(a*b,u) a,b,u≧0

//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; }

inline 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);
  }
  template<class T,size_t N>
  static void cin1core(array<T,N> &a){for(int i=0;i<(int)N;++i) cin>>a[i];}
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...>>(); }


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 ----*/
  constexpr mll_(){}
  mll_(ll v): val_(v){ norm(); }
  constexpr 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--(int){ return *this-=1; }
  mll_ &operator++(int){ return *this+=1; }
  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 };
};


struct bll{
  ll s=0;
  bll(ll s_=0): s(s_){}
  bll(int s_): s(s_){}
  bll(const string &bitstr): s(str2val(bitstr)){}
  bll(const char   *bitstr): s(str2val(bitstr)){}
  struct ref {
    bll &b; const ll msk;
    ref(bll &b_,ll pos):b(b_),msk(1LL<<pos){}
    operator ll() const { return (b.s&msk)!=0; }
    ref &operator=(bool x){ if(x) b.s|=msk; else b.s&=~msk; return *this; }
  };
  ref operator[](ll pos){ return ref(*this,pos); }
  ll operator[](ll pos) const { return (s>>pos)&1; }
  bll &operator=(int b){ s=b; return *this; }
  bll &operator=(ll  b){ s=b; return *this; }
  bll &operator=(const string &bitstr){ s=str2val(bitstr); return *this; }
  bll &operator=(const char   *bitstr){ s=str2val(bitstr); return *this; }
  bll operator++(int){ bll b(*this); s++; return b; }
  bll operator--(int){ bll b(*this); s--; return b; }
  operator ll() const noexcept { return s; }
  bll &operator&=(ll b){ s&=b; return *this; }
  bll &operator|=(ll b){ s|=b; return *this; }
  bll &operator^=(ll b){ s^=b; return *this; }
  bll &operator+=(ll b){ s+=b; return *this; }
  bll &operator-=(ll b){ s-=b; return *this; }
  bll &operator<<=(ll i){ s<<=i; return *this; }
  bll &operator>>=(ll i){ s>>=i; return *this; }
  bll operator&(ll b)const{ return s&b; }
  bll operator|(ll b)const{ return s|b; }
  bll operator^(ll b)const{ return s^b; }
  bll operator+(ll b)const{ return s+b; }
  bll operator-(ll b)const{ return s-b; }
  bll operator<<(ll i)const{ return s<<i; }
  bll operator>>(ll i)const{ return s>>i; }
  bll operator&(int b)const{ return s&b; }
  bll operator|(int b)const{ return s|b; }
  bll operator^(int b)const{ return s^b; }
  bll operator+(int b)const{ return s+b; }
  bll operator-(int b)const{ return s-b; }
  bll operator<<(int i)const{ return s<<i; }
  bll operator>>(int i)const{ return s>>i; }
  bll operator~()const{ return ~s; }
  bll &oneq  (bll msk){ s|= msk.s; return *this; }
  bll &offeq (bll msk){ s&=~msk.s; return *this; }
  bll &flipeq(bll msk){ s^= msk.s; return *this; }
  bll on   (bll msk)const{ return bll(s).oneq  (msk); }
  bll off  (bll msk)const{ return bll(s).offeq (msk); }
  bll flip (bll msk)const{ return bll(s).flipeq(msk); }
  bool any0(bll msk)const{ return ~s&msk.s; }
  bool any1(bll msk)const{ return  s&msk.s; }
  bool all0(bll msk)const{ return !any1(msk); }
  bool all1(bll msk)const{ return !any0(msk); }
  bll &oneq  (ll l,ll r){ return oneq  (rngmsk(l,r)); }
  bll &offeq (ll l,ll r){ return offeq (rngmsk(l,r)); }
  bll &flipeq(ll l,ll r){ return flipeq(rngmsk(l,r)); }
  bll on   (ll l,ll r)const{ return on  (rngmsk(l,r)); }
  bll off  (ll l,ll r)const{ return off (rngmsk(l,r)); }
  bll flip (ll l,ll r)const{ return flip(rngmsk(l,r)); }
  bool any0(ll l,ll r)const{ return any0(rngmsk(l,r)); }
  bool any1(ll l,ll r)const{ return any1(rngmsk(l,r)); }
  bool all0(ll l,ll r)const{ return all0(rngmsk(l,r)); }
  bool all1(ll l,ll r)const{ return all1(rngmsk(l,r)); }
  bll &maskeq(ll l,ll r){ s&=rngmsk(l,r); return *this; }
  bll mask(ll l,ll r)const{ return bll(s).maskeq(l,r); }
  bll &oneq  (ll i){ s|= (1LL<<i); return *this; }
  bll &offeq (ll i){ s&=~(1LL<<i); return *this; }
  bll &flipeq(ll i){ s^= (1LL<<i); return *this; }
  bll on  (ll i)const{ return s| (1LL<<i); }
  bll off (ll i)const{ return s&~(1LL<<i); }
  bll flip(ll i)const{ return s^ (1LL<<i); }
  bool contains(ll b)const{ return (s&b)==b; }
  bll substr(ll l,ll r)const{ return (s&rngmsk(l,r))>>r; }
  static bll rngmsk(ll l,ll r){ return (1LL<<(l+1))-(1LL<<r); }
  ll msbit()const{
    for(ll x=63,o=-1;;){
      ll m=(x+o)/2;
      if((1LL<<m)<=s) o=m; else x=m;
      if(x-o==1) return o;
    }
  }
  ll lsbit()const{ return bll(lsb()).msbit(); }
  ll msb()const{ ll pos=msbit(); return (pos<0) ? 0LL : 1LL<<pos; }
  ll lsb()const{ return s&-s; }
  ll count()const{ return bitset<64>(s).count(); }
  ll count(bll msk)const{ return (msk&s).count(); }
  ll count(ll l,ll r)const{ return mask(l,r).count(); }
  vector<ll> idxes()const{
    vector<ll> v;
    for(ll i=0,t=s; t; t>>=1,i++) if(t&1)v.push_back(i);
    return v;
  }
  string to_string(ll wd=-1)const{
    wd=max({wd,msbit()+1,1LL});
    string ret;
    for(ll i=wd-1;i>=0;--i) ret += '0'+char((s>>i)&1);
    return ret;
  }
private:
  ll str2val(const string &bitstr){
    ll val=0, len=(ll)bitstr.size();
    for(ll i=0;i<len;++i) val|=ll(bitstr[i]-'0')<<(len-1-i);
    return val;
  }
};


template<class T> struct SET: set<T>{
  using P=set<T>;
  typename P::iterator it=P::end();
  template<class...Args> SET(Args...args): P(args...){}
  SET(initializer_list<T> a): P(a.begin(),a.end()){}
  ll size() const { return (ll)P::size(); }
  bool insert(const T &x){ bool r; tie(it,r)=P::insert(x); return r; }
  template <class It> void insert(It st,It en){ P::insert(st,en); }
  void insert(initializer_list<T> a){ P::insert(a.begin(),a.end()); }
  template<class...A> bool emplace(A&&...a){ bool r; tie(it,r)=P::emplace(a...); return r; }
  void eraseit(){ it=P::erase(it); }
  void find(const T &x){ it=P::find(x); }
  bool contains(const T &x){ return P::count(x)==1; }
  void lower_bound(const T &x){ it=P::lower_bound(x); }
  void upper_bound(const T &x){ it=P::upper_bound(x); }
  bool isend() { return it==P::end(); }
  T getit() { return *it; }
  T next() { return *(++it); }
  T prev() { return *(--it); }
  bool nextok() { return !isend() && it!=--P::end(); }
  bool prevok() { return it!=P::begin(); }
  T front() { return *(it=P::begin()); }
  T back()  { return *(it=--P::end()); }
  void pop_front(){ front(); eraseit(); }
  void pop_back(){ back(); eraseit(); }
  void push_front(const T &x){ it=P::insert(P::begin(),x); }
  void push_back (const T &x){ it=P::insert(P::end(),x); }
  void push_out(SET &b){ b.push_front(back()); pop_back(); }
  void pull_in(SET &b){ push_back(b.front()); b.pop_front(); }
};


template<class T> struct cumulativesum{
  using Int = long long;
  using ll = long long;
  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()(Int l,Int r){ return c[max(min(n,r+1),0LL)]-c[min(max(0LL,l),n)]; }
  pair<Int,T> group(T i){
    ll g=upper_bound(c.begin(),c.end(),i)-c.begin()-1;
    T r = g>=0 ? i-c[g] : i;
    return {g,r};
  }
  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>;


template<class T> vector<T> powers(T m,ll n){
  vector<T> ret(n+1,1);
  for(ll i=1;i<=n;++i) ret[i]=ret[i-1]*m;
  return ret;
}


template <class T> auto runlength(T &&v){
  vector<pair<typename remove_reference<T>::type::value_type,ll>> ret;
  for(auto&&e:v){
    if(ret.empty() or ret.back().first!=e) ret.emplace_back(e,1);
    else ret.back().second++;
  }
  return ret;
}


inline vector<ll> str2num(string &s,char base,const string &etc){
  vector<ll>  v(s.size());
  for(ll i=0;i<(ll)s.size();++i){
    size_t pos=etc.find(s[i]);
    if(pos==etc.npos) v[i]=s[i]-(ll)base;
    else v[i]=-((ll)pos+1);
  }
  return v;
}


template<class T> struct combination{
  vector<T> f,g; ll mxN=0;
  combination(){}
  combination(ll maxN): f(maxN+1,1),g(maxN+1),mxN(maxN) {
    for (ll i=1;i<=mxN;++i) { f[i]=f[i-1]*i; }
    g[mxN]=1/f[mxN];
    for (ll i=mxN;i>=1;--i) { g[i-1]=g[i]*i; }
  }
  T P(ll n,ll r){ return (n<0 || r<0 || n<r) ? T(0) : f[n]*g[n-r]; } //nPr
  T H(ll n,ll r){ return operator()(n+r-1,n-1); }//nHr
  T inv(ll n) { return f[n-1] * g[n]; } //1/n
  T fact(ll n) { return f[n]; } //n!
  T finv(ll n) { return g[n]; } //1/n!
  T operator()(ll n,ll r){
    if (r<0) return 0;
    if (n<0) return operator()(-n+r-1,r) * ((r&1)?-1:1); //-nＣr = (-1)^r * n+r-1Ｃr
    if (n<r) return 0;
    if (n<=mxN) return f[n]*g[n-r]*g[r]; //通常
    //n巨大、rかn-r小
    if (n-r<r) r=n-r;
    T bunsi=1,bunbo=1;
    for (ll i=0;i<r;++i) bunsi*=n-i;
    for (ll i=0;i<r;++i) bunbo*=i+1;
    return bunsi/bunbo;
  }
  template<class SP>
  vector<T> CnLnR(long long nL,long long nR,long long r,SP sp){
    if (nR-nL+1<=0) return vector<T>();
    if (r<0) return vector<T>(nR-nL+1,0);
    vector<T> v=sp(nL-r+1,nR-r+1,r);
    for (T& e: v) e*=finv(r);
    return v;
  }
  template<class SP>
  vector<T> HrLrR(long long n,long long rL,long long rR,SP sp){//r<0不可
    return CnLnR(n-1+rL,n-1+rR,n-1,sp);
  }
};


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);
  }
};


namespace dumpstring{//dummy
inline string stringf(const char *format,...){
  char bf[1000];
  va_list ap;
  va_start(ap,format);
  vsprintf(bf,format,ap);
  va_end(ap);
  return string(bf);
}
template <class T> string stringfx(T x,int wd=1){ return ""; }
struct args{
  using Int = long long;
  args(){}
  args &wd(Int wd__){ (void)wd__; return *this; }
  args &sx(Int s){ (void)s; return *this; }
  template<size_t DIM> args &rngs(array<array<Int,DIM>,2> rngs){ return *this; }
  args &tr(vector<Int> tr__){ (void)tr__; return *this; }
  args &tr(){ return *this; }
  args &labels(vector<string> labels__){ (void)labels__; return *this; }
  args &xrev(){ return *this; }
  args &yrev(){ return *this; }
  args &zrev(){ return *this; }
  args &wrev(){ return *this; }
};
template<class NdT>
void dumpNd(const string &h,const NdT &fd,const args &p=args(),ostream &os=cerr){}
};
using dumpstring::stringf; using dumpstring::stringfx;
using dumpstring::args; using dumpstring::dumpNd;

#endif//テンプレートend


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)){}
  Vector(int n,cT& x):vT(n,x){}// ┬ 型推論のためラッパー
  Vector(long long n,cT& x):vT(n,x){}
  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); }
  Vector &operator+=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]+=b[i]; return *this; }
  Vector &operator-=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]-=b[i]; return *this; }
  Vector &operator*=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]*=b[i]; return *this; }
  Vector &operator/=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]/=b[i]; return *this; }
  Vector &operator%=(cvT &b){ assert(size()==b.size()); for(Int i=0;i<size();++i) (*this)[i]%=b[i]; return *this; }
  Vector &operator+=(const Vector<T> &b){ return *this+=(cvT&)b; }
  Vector &operator-=(const Vector<T> &b){ return *this-=(cvT&)b; }
  Vector &operator*=(const Vector<T> &b){ return *this*=(cvT&)b; }
  Vector &operator/=(const Vector<T> &b){ return *this/=(cvT&)b; }
  Vector &operator%=(const Vector<T> &b){ return *this%=(cvT&)b; }
  Vector operator+(cvT &b){ return Vector(*this)+=b; }
  Vector operator-(cvT &b){ return Vector(*this)-=b; }
  Vector operator*(cvT &b){ return Vector(*this)*=b; }
  Vector operator/(cvT &b){ return Vector(*this)/=b; }
  Vector operator%(cvT &b){ return Vector(*this)%=b; }
  Vector operator+(const Vector<T> &b){ return Vector(*this)+=b; }
  Vector operator-(const Vector<T> &b){ return Vector(*this)-=b; }
  Vector operator*(const Vector<T> &b){ return Vector(*this)*=b; }
  Vector operator/(const Vector<T> &b){ return Vector(*this)/=b; }
  Vector operator%(const Vector<T> &b){ return Vector(*this)%=b; }
  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--(int){ return *this-=1; }
  Vector &operator++(int){ return *this+=1; }
  Vector operator-()const{ return Vector(*this)*=-1; }
  template<class S> friend Vector operator-(S x,const Vector &a){ return -a+=x; }
  Vector slice(Int l,Int r,Int d=1)const{
    Vector ret;
    for(Int i=l;(d>0&&i<=r)||(d<0&&r<=i);i+=d) ret.push_back((*this)[i]);
    return ret;
  }
  Int size()const{ return (Int)vT::size(); }
  Vector &push_back(cT& x,Int n=1){ for(Int i=0;i<n;++i){ vT::push_back(x); } return *this; }
  Vector &pop_back(Int n=1){ for(Int i=0;i<n;++i){ vT::pop_back(); } return *this; }
  Vector &push_front(cT& x,Int n=1){ this->insert(0,x,n); return *this; }
  Vector &pop_front(Int n=1){ erase(0,n-1); return *this; }
  T pull_back(){ T x=move(vT::back()); vT::pop_back(); return x; }
  T pull_front(){ T x=move(vT::front()); erase(0); return x; }
  Vector &insert(Int i,cT& x,Int n=1){ insert(it(i),n,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,Int n=1){
    cvT B = (&b==this) ? *this : vT{};
    for(int i=0;i<n;++i) this->insert(size(),(&b==this)?B:b);
    return *this;
  }
  Vector repeat(Int n){ return Vector{}.concat(*this,n); }
  Vector &reverse(Int l=0,Int r=-1){ r+=r<0?size():0; std::reverse(it(l),it(r+1)); return *this; }
  Vector &rotate(Int m){ return rotate(0,size()-1,m); }
  Vector &rotate(Int l,Int r,Int m){ std::rotate(it(l),it(m),it(r+1)); return *this; }
  Vector &sort(Int l=0,Int r=-1){ r+=r<0?size():0; std::sort(it(l),it(r+1)); return *this; }
  Vector &rsort(Int l=0,Int r=-1){ return sort(l,r).reverse(l,r); }
  template<class Pr> Vector &sort(Pr pr){ return sort(0,size()-1,pr); }
  template<class Pr> Vector &sort(Int l,Int r,Pr pr){ std::sort(it(l),it(r+1),pr); return *this; }
  template<int key> Vector &sortbykey(Int l=0,Int r=-1){
    r+=r<0?size():0;
    sort(l,r,[](cT &x,cT &y){return get<key>(x)<get<key>(y);});
    return *this;
  }
  Vector &uniq(){ erase(unique(begin(),end()),end()); return *this; }
  Vector &sortq(){ return sort().uniq(); }
  Vector &fill(cT& x){ return fill(0,size()-1,x); }
  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 count(0,size()-1,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 countif(0,size()-1,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 find(0,size()-1,x); }
  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 findif(0,size()-1,pr); }
  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; }

  template<class S> using VV    = Vector<Vector<S>>; template<class S> using sVV    = vector<vector<S>>;
  template<class S> using VVV   = Vector<VV<S>>;     template<class S> using sVVV   = vector<sVV<S>>;
  template<class S> using VVVV  = Vector<VVV<S>>;    template<class S> using sVVVV  = vector<sVVV<S>>;
  template<class S> using VVVVV = Vector<VVVV<S>>;   template<class S> using sVVVVV = vector<sVVVV<S>>;
  auto tostd()const{ return tov(*this); }
  template <class S> static vector<S> tov(const Vector<S>&v){ return v; }
  template <class S> static sVV<S>    tov(const VV<S>    &v){ sVV<S>    ret; for(auto&& e:v) ret.push_back(e);         return ret; }
  template <class S> static sVVV<S>   tov(const VVV<S>   &v){ sVVV<S>   ret; for(auto&& e:v) ret.push_back(e.tostd()); return ret; }
  template <class S> static sVVVV<S>  tov(const VVVV<S>  &v){ sVVVV<S>  ret; for(auto&& e:v) ret.push_back(e.tostd()); return ret; }
  template <class S> static sVVVVV<S> tov(const VVVVV<S> &v){ sVVVVV<S> ret; for(auto&& e:v) ret.push_back(e.tostd()); return ret; }
};
/*
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 a++ a--  //∀i a[i]にxを演算
a+=b a-=b a*=b a/=b a%=b a+b a-b a*b a/b a%b  //要素毎演算(同サイズ時)
a.push_front(x,n); //n個先頭追加 省略時1
a.push_back(x,n);  //n個末尾追加 省略時1
a.pop_front(n);    //n個先頭削除 省略時1
a.pop_back(n);     //n個末尾削除 省略時1
ll x=a.pull_front(); //pop_front()と同時に値取得
ll x=a.pull_back();  //pop_back()と同時に値取得
a.insert(i,x,n); //a[i]にn個x挿入  n省略時1
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 b=a可
a.concat(b,n);   //aにbをn回結合 b=a可
a.reverse(l,r);  //[l,r]を反転       l,r省略可
a.rotate(m);     //a[m]を先頭にするrotate
a.rotate(l,r,m); //a[m]を先頭にするrotate 範囲[l,r]
a.sort(l,r);     //[l,r]をソート     l,r省略可
a.rsort(l,r);    //[l,r]を逆順ソート l,r省略可
p.sort(l,r,[&](pll x,pll y){return x.second<y.second;});//比較関数指定sort l,r省略可
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        l,r省略可
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
vll b=a.slice(st,en,d); //a[st:en:d]   d省略時1
vll b=a.repeat(n);   //aをn回繰り返す
- -------- 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
auto aa=a.tostd(); //N次元VectorをN次元vectorに変換(N≦5)
*/


template<class T> struct wrapv: Vector<T>{
  using Int = long long;
  T def=T();
  T defIF=T();
  wrapv(const Vector<T> &b):Vector<T>(b){}
  wrapv(Vector<T> &&b):Vector<T>(move(b)){}
  wrapv(const std::vector<T> &b):Vector<T>(b){}
  wrapv(std::vector<T> &&b):Vector<T>(move(b)){}
  T &operator[](Int i){
    return (i<0 || this->size()<=i) ? (defIF=def) : Vector<T>::operator[](i);
  }
  void setdef(const T& x){ def=x; }
};
/*
wrapv v=vll(N,0,1); //vllなどでコンストラクトしてから代入する
v.setdef(INF); //範囲外での値セット
*/


#if 0
#define MODLL (1000000007LL)
#else
#define MODLL (998244353LL)
#endif
using mll = mll_<MODLL>;
//using mll = fraction;

//　１
//０┼２
//　３              左     上     右    下
vector<pll> dxys={{0,-1},{-1,0},{0,1},{1,0},};



#ifdef _MSC_VER
struct ll128{
  using ull=unsigned long long;
  int sign=1; //+1:非負、-1:負
  ull upper=0,lower=0; //上位64bit、下位64bit
  ll128(){}
  template<integral T> ll128(T x):sign(x>=0 ? 1 : -1),lower(ull(abs(x))){}
  operator ll()const{ assert(upper==0 && (lower>>63)==0); return sign*ll(lower); }
  template<integral T> ll128 operator+(T b)const{ return *this+ll128(b); }
  template<integral T> ll128 operator-(T b)const{ return *this-ll128(b); }
  template<integral T> ll128 operator*(T b)const{ return *this*ll128(b); }
  template<integral T> ll128 operator/(T b)const{ return *this/ll128(b); }
  template<integral T> ll128 operator%(T b)const{ return *this%ll128(b); }
  template<integral T> ll128 operator+=(T b){ return *this+=ll128(b); }
  template<integral T> ll128 operator-=(T b){ return *this-=ll128(b); }
  template<integral T> ll128 operator*=(T b){ return *this*=ll128(b); }
  template<integral T> ll128 operator/=(T b){ return *this/=ll128(b); }
  template<integral T> ll128 operator%=(T b){ return *this%=ll128(b); }
  template<integral T> friend ll128 operator+(T a,ll128 b){ return b+a; }
  template<integral T> friend ll128 operator-(T a,ll128 b){ return -b+a; }
  template<integral T> friend ll128 operator*(T a,ll128 b){ return b*a; }
  template<integral T> friend ll128 operator/(T a,ll128 b){ return ll128(a)/b; }
  ll128 operator+(ll128 b)const{ return ll128(*this)+=b; }
  ll128 operator-(ll128 b)const{ return ll128(*this)-=b; }
  ll128 operator*(ll128 b)const{ return ll128(*this)*=b; }
  ll128 operator/(ll128 b)const{ return ll128(*this)/=b; }
  ll128 operator%(ll128 b)const{ return ll128(*this)%=b; }
  ll128 operator-()const{ ll128 ret(*this); ret.sign*=-1; return ret; }
  ll128 &operator+=(ll128 b){
    //同符号なら中身加算、異符号なら中身引き算
    if(sign==b.sign)          tie(upper,lower)=addNoSign(*this,b);
    else if(geNoSign(*this,b)) tie(upper,lower)=subNoSign(*this,b);
    else                       tie(upper,lower)=subNoSign(b,*this),sign=-sign;
    return *this;
  }
  ll128 &operator-=(ll128 b){
    return (*this)+=(-b);
  }
  ll128 &operator*=(ll128 b){
    sign*=b.sign;
    ull ansu,ansl;
    ansl=_umul128(lower,b.lower,&ansu);
    //以下overflowしない前提で計算
    ansu+=upper*b.lower;
    ansu+=lower*b.upper;
    tie(upper,lower)={ansu,ansl};
    return *this;
  }
  ll128 &operator/=(ll128 b){//bが64bitに収まる前提で計算 負の扱いは適当
    sign*=b.sign;
    ull q = _udiv128(upper,lower,b.lower,nullptr);
    upper=0,lower=q;
    return *this;
  }
  ll128 &operator%=(ll128 b){//bが64bitに収まる前提で計算 負の扱いは適当
    sign*=b.sign;
    ull rem;
    _udiv128(upper,lower,b.lower,&rem);
    upper=0,lower=rem;
    return *this;
  }
  bool operator==(ll128 b)const{
    return sign==b.sign && upper==b.upper && lower==b.lower;
  }
  bool operator!=(ll128 b)const{ return !(*this==b); }
  bool operator<(ll128 b)const{
    return sign<b.sign || //負と正
      (sign==1  && b.sign==1  && !geNoSign(*this,b)) || //正と正
      (sign==-1 && b.sign==-1 && !geNoSign(b,*this));   //負と負
  }
  bool operator> (ll128 b)const{ return b<*this; }
  bool operator>=(ll128 b)const{ return !(*this<b); }
  bool operator<=(ll128 b)const{ return b>=*this; }
private:
  static pair<ull,ull> addNoSign(ll128 a,ll128 b){//符号無視で加算
    ull u=a.upper,l=a.lower,u2=b.upper,l2=b.lower;
    return {u+u2+(l>~l2),l+l2}; //overflow判定 l+l2>all1 → l>all1-l2=~l2
  }
  static pair<ull,ull> subNoSign(ll128 a,ll128 b){//符号無視,a≧b前提で加算
    ull u=a.upper,l=a.lower,u2=b.upper,l2=b.lower;
    return {u-u2-(l<l2),l-l2};
  }
  static bool geNoSign(ll128 a,ll128 b){//true: 符号無視でa≧b
    ull u=a.upper,l=a.lower,u2=b.upper,l2=b.lower;
    return u>u2 || (u==u2 && l>=l2);
  }
};
#else
using ll128=__int128_t;
#endif


template<long long MOD,class LL128=ll128> struct static_mll128{
  using MLL=static_mll128;
  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(ll a){//aとMODが互いに素の時のみ
    ll b=MOD,u=1,v=0;
    while(b){
      ll t=a/b;
      a-=t*b, swap(a,b);
      u-=t*v, swap(u,v);
    }
    u %= MOD;
    if (u<0) u+=MOD;
    return u;
  }
  /*---- I/F ----*/
  constexpr static_mll128(){}
  static_mll128(ll v): val_(v){ norm(); }
  constexpr static_mll128(ll v,bool b): val_(v){} //正規化無のコンストラクタ
  ll val()const{ return 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(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_=LL128(val_)*b.val_%MOD; return *this; }
  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--(int){ --val_; return normS(); }
  MLL &operator++(int){ ++val_; return normP(); }
  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 ll mod() { return MOD; }
};
/*
MOD<2^62まで使えるmll
templateに128bit整数型を指定する
using mll128 = static_mll128<(1LL<<61)-1,int128_t>;
mll128 a;
*/


template<int id,class LL128=ll128> struct dynamic_mll128{
  using MLL=dynamic_mll128;
  using ll=long long;
  ll val_=0;
  static inline ll MOD=998244353;
  /*---- 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(ll a){//aとMODが互いに素の時のみ
    ll b=MOD,u=1,v=0;
    while(b){
      ll t=a/b;
      a-=t*b, swap(a,b);
      u-=t*v, swap(u,v);
    }
    u %= MOD;
    if (u<0) u+=MOD;
    return u;
  }
  /*---- I/F ----*/
  constexpr dynamic_mll128(){}
  dynamic_mll128(ll v): val_(v){ norm(); }
  constexpr dynamic_mll128(ll v,bool b): val_(v){} //正規化無のコンストラクタ
  static void set_mod(ll m){ assert(1<=m); MOD=m; }
  ll val()const{ return 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(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_=LL128(val_)*b.val_%MOD; return *this; }
  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--(int){ --val_; return normS(); }
  MLL &operator++(int){ ++val_; return normP(); }
  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 pow128(ll128 k)const{
    MLL ret(1,false),a(*this);
    for(; k>ll128(0); k/=2,a*=a) if(k&1)ret*=a;
    return ret;
  }
  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 ll mod() { return MOD; }
};
/*
MOD<2^62まで使えるmll、動的MOD版
templateに128bit整数型を指定する
using mll128 = dynamic_mll128<0,int128_t>;
mll128 a;
*/


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 vvvvdd=vector<vvvdd>;
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   vll2=vector<  ll2>; using    vll3=vector<   ll3>; using   vll4=vector<  ll4>;
using  vvll2=vector< vll2>; using   vvll3=vector<  vll3>; using  vvll4=vector< vll4>;
using vvvll2=vector<vvll2>; using  vvvll3=vector< vvll3>; using vvvll4=vector<vvll4>;
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); }); }


struct rndutil64{
  using Int = long long;
  using  ll = long long;
  using ull = unsigned long long;
  mt19937_64 mt;
  rndutil64(ll seed=0):mt((ull)seed){}
  ll val(ll a,ll b){ return ll(mt()%(b-a+1))+a; }//[a,b]の乱数
  vector<ll> ary(Int n,ll a,ll b){//長さnの配列、要素[a,b]内、重複可
    vector<ll> v(n);
    for(ll i=0;i<n;++i) v[i]=this->val(a,b);
    return v;
  }
  vector<ll> sample(Int n,ll a,ll b){//長さnの配列、要素[a,b]内、重複無
    ll len=b-a+1;
    assert((ll)n<=len);
    vector<ll> v(len);
    iota(v.begin(),v.end(),a);
    for(ll i=0;i<n;++i) swap(v[i],v[this->val(i,len-1)]);
    v.resize(n);
    return v;
  }
private:
  ull getseed(ll seed){
    if(seed>=0) return (ull)seed;
    return (ull)chrono::system_clock::now().time_since_epoch().count();
  }
};
/*
rndutil64 r64;
- -------- 整数x -------- 区間[a,b]
ll x=r64.val(a,b);
- -------- 整数ベクトルv -------- 長さn,要素の区間[a,b] 配列重複有
vll v=r64.ary(n,a,b);
- -------- 整数ベクトルv -------- 長さn,要素の区間[a,b] 配列重複無
vll v=r64.sample(n,a,b);
*/



#ifdef _MSC_VER
#include <intrin.h>
#endif
struct primefactorization{
  using ull = unsigned long long;
  ll fmax=0; //篩のmax
  vll mnpr; //最小素因数表
  vll primes; //素数リスト
  vpll pfact; //素因数分解結果 例:{<2,3>,<3,1>,<5,2>}→2^3×3^1×5^2
  vll divs; //約数列挙結果
  primefactorization(){} //篩無し
  primefactorization(ll fmx){ init(fmx); } //篩使用
  void init(ll fmx){ //エラトステネスの篩で最小素因数表+素数リストを作成
    fmax=fmx;
    mnpr.resize(fmx+1);
    primes.reserve(fmx/10+1);
    for(ll i=2; i<=fmax; i++){
      if(mnpr[i]) continue;
      primes.push_back(i);
      mnpr[i]=i;
      for(ll j=i*i; j<=fmax; j+=i) if(mnpr[j]==0) mnpr[j]=i;
    }
  }
  vll &primelist(){ return primes; }
  bool isprime(ll a){
    if(a<=fmax) return mnpr[a]==a;
    else return MillerRabin((ull)a);
  }
  vpll &operator()(ll a){//素因数分解
    return this->pfact=Factorization(a);
  }
  vll pfactlist(ll a){//素因数リスト(指数は返さない)
    (*this)(a);//素因数分解実行
    vll ret;
    for(auto&&[p,_]:this->pfact) ret.push_back(p);
    return ret;
  }
  /*!
  @brief   素因数分解結果pfactから約数列挙divsを得る
  @details e乗しても約数のもののみ raise=trueならe乗後を返す
  */
  static void divisorCore(const vpll &pfact,vll &divs,ll e=1,bool raise=false){
    divs.assign(1,1);
    for(auto [p,nm]: pfact){
      ll prenm=(ll)divs.size();
      for(int i=0;i<prenm*(nm/e);++i) divs.push_back(divs[i]*p);
    }
    if(raise){ //e乗
      for(auto&& y: divs){
        for(ll i=1,yorg=y;i<e;++i) y*=yorg;
      }
    }
  }
  vll &divisor(ll a,ll e=1,bool raise=false){//約数列挙 e乗して約数のもの
    (*this)(a);//素因数分解実行
    divisorCore(this->pfact,this->divs,e,raise);//約数列挙コア
    return divs;
  }
  vpll Factorization(ll a){
    if(a<(1LL<<31)) return FactorizationCore((int)a);
    else             return FactorizationCore(a);
  }
  template<class T> vpll FactorizationCore(T a){//素因数分解
    vpll ret;
    if(a<=1) return ret;
    if(a<=fmax){//篩の範囲→osa_k法
      for(; mnpr[a]!=a; a/=(T)mnpr[a]) Add(ret,mnpr[a]);
      Add(ret,a);
      return ret;
    }
    if(MillerRabin((ull)a)){//素数の時
      ret.emplace_back(a,1);
      return ret;
    }
    //それ以外→ロー法で再帰
    ll y=RhoAlgorithm((ull)a);
    vpll ret1=Factorization(y);
    vpll ret2=Factorization(a/y);
    ll i=0,j=0,len1=(ll)ret1.size(),len2=(ll)ret2.size();
    while(i<len1 || j<len2){
      if(j==len2 || (i<len1 && ret1[i]<ret2[j])) AddBlock(ret,ret1[i++]);
      else                                        AddBlock(ret,ret2[j++]);
    }
    return ret;
  }
  bool MillerRabin(ull n){//true:素数 定数は http://miller-rabin.appspot.com/ より
    auto modpow=[&](ull a,ull b,ull m){ //a^b (mod m)
      ull r=1;
      for(; b>0; b>>=1,a=ModMul(a,a,m)) if(b&1) r=ModMul(r,a,m);
      return r;
    };
    auto f=[&](ull a){//true:素数かもしれない、false:合成数確定
      a%=n;
      if(a==0) return true;
      ull t=n-1,s=0;
      while(t%2==0) t>>=1,s++;
      ull x=modpow(a,t,n);
      if(x==1 || x==n-1) return true;
      for(ull r=1; r<s; ++r){
        if((x=ModMul(x,x,n))==n-1) return true;
      }
      return false;
    };
    if(n<=1) return false;
    if(n==2) return true;
    if(n%2==0) return false;
    if(n<4759123141) return f(2)&&f(7)&&f(61);
    return f(2)&&f(325)&&f(9375)&&f(28178)&&f(450775)&&f(9780504)&&f(1795265022);
  }
  ll RhoAlgorithm(ull n){//約数の1つを返す 素数はNG(無限ループする)
    if((n&1)==0) return 2;
    for(ull c=1,x=1;;++c){
      auto f=[&](ull x){ return (ModMul(x,x,n)+c)%n; };
      ull y=x,g=1;
      while(g==1){
        x=f(x);
        y=f(f(y));
        g=gcd(abs(ll(x-y)),n);
      }
      if(g<n) return (ll)g;
    }
  }
  void AddBlock(vpll &v,pll &p){
    if(!v.empty() and v.back().first==p.first) v.back().second+=p.second;
    else v.push_back(p);
  }
  void Add(vpll &v,ll x){
    if(!v.empty() and v.back().first==x) v.back().second++;
    else v.emplace_back(x,1);
  }
  inline ull ModMul(ull a,ull b,ull m){ //a*b%m
    if(a < 1ULL<<32 && b < 1ULL<<32) return a*b%m;
#ifdef _MSC_VER
    ull upper,lower,rem;
    lower=_umul128(a,b,&upper);
    _udiv128(upper,lower,m,&rem);
    return rem;
#else
    return (ull)((unsigned __int128)(a)*b%m);
#endif
  }
};
/*
- -------- 定義 -------- M:篩最大値
primefactorization pr;
primefactorization pr(M);
- -------- 素数判定 -------- 篩内なら使用、篩外ならミラーラビンで判定
bool b=pr.isprime(x);
- -------- 素数リスト --------  {2,3,5,7,11,…}
vll &primes=pr.primelist();
- -------- 素因数分解 --------
vpll &pfact=pr(x);
.      ↑例: x=1  → {}
.            x=2  → {<2,1>} //2^1の意味
.            x=600→ {<2,3>,<3,1>,<5,2>} //2^3×3^1×5^2 素因数昇順
- -------- 素因数分解(指数なし) -------- 例:x=600→{2,3,5} (600=2^3×3^1×5^2)
vll plist=pr.pfactlist(x);
- -------- 約数列挙 -------- 例:x=12→{1,2,4,3,6,12}＜未ソート＞
vll &div=pr.divisor(x);
vll &div=pr.divisor(x,e);      //e乗が約数のものを列挙
vll &div=pr.divisor(x,e,true); //約数でe乗数のものを列挙
*/


template<class T> ll modpow(ll a,ll x,ll M){
  ll r=1;
  for(; x>0; x>>=1,a=T(a)*a%M) if(x&1) r=T(r)*a%M;
  return r;
}
/*
- ---- a^x mod M ----
ll b=modpow<ll>   (a,x,M);  //a と M が31bit以下のとき
ll b=modpow<ll128>(a,x,M);  //a or M が32bit以上のとき
*/


template<class T,class Func>
ll getOrder(ll a,ll m,const vector<ll> &plist,Func getPFactList){
  assert(m>=2);
  //a^φ(m)=1(mod m)なのでφ(m)の約数を調べる
  ll phi=m;
  for(auto&& p: plist){
    assert(a%p!=0); //gcd(a,m)==1が前提
    phi=phi/p*(p-1);
  }
  vector<ll> plistOfPhi=getPFactList(phi); //φ(m)の素因数リスト作成
  ll ord=phi;
  for(auto&& p: plistOfPhi){
    while(ord%p==0 && modpow<T>(a,ord/p,m)==1) ord/=p;
  }
  return ord;
}
/*
- ---- 法mでのaの位数 (a^x≡1(mod m)なる最小のx) ----
auto getPFactList=[&](ll x){ return primefactorization().pfactlist(x); };
ll x=getOrder<ll>   (a,m, getPFactList(m), getPFactList); //a と m が31bit以下のとき
ll x=getOrder<ll128>(a,m, getPFactList(m), getPFactList); //a or m が32bit以上のとき
.        mの素因数リスト(指数は不要)↑             ↑mの素因数リスト作成関数
*/


void cin2solve()
{
  auto K=cin1<ll>();
  auto ST=cinv<pair<string,ll>>(K);

  primefactorization pr;
  rndutil64 r64(-1);

  /*
  64bitMODで安全にし、MODランダムにして落とされないようにする
  */
  using mll128=dynamic_mll128<0>;
  ll mod;
  do{
    mod=r64.val(1000000000000000001,2000000000000000001);
  }
  while(!pr.isprime(mod));
  mll128::set_mod(mod);

  ll base;//baseもランダム化
  while(true){
    base=r64.val(100000,mod-100000);
    ll ord=getOrder<ll128>(base,mod,pr.pfactlist(mod),[&](ll x){ return pr.pfactlist(x); });
    if(ord>300000) break;//分母が0にならないよう位数を文字列長以上にする
  }

  using Hash=pair<mll128,ll128>;
  auto calcstringhash=[&](string &s)->Hash{
    mll128 h=0;
    for(auto&& c: s) h=h*base+c;
    return {h,sz(s)};
  };

#if 0
  auto hashrepeat=[&](Hash h,ll r){
    auto [ha,n]=h;
    mll128 B=base;
    mll128 bunsi=B.pow128(n*r)-1;
    mll128 bunbo=B.pow(n)-1;
    return Hash{ha*bunsi/bunbo,n*r};
  };

  auto hashconcat=[&](Hash h1,Hash h2)->Hash{
    auto [ha1,n1]=h1;
    auto [ha2,n2]=h2;
    mll128 ha3=ha1*mll128(base).pow128(n2) + ha2;
    return {ha3,n1+n2};
  };
#endif

  auto repeatconcat=[&](Hash h1,Hash h2,ll r){
    auto [ha1,n1]=h1;
    auto [ha2,n2]=h2;
    mll128 B=base;
    mll128 Bn2=B.pow(n2);
    mll128 Bn2r=Bn2.pow(r);
    mll128 bunsi=Bn2r-1;
    mll128 bunbo=Bn2-1;
    mll128 Brep=ha2*bunsi/bunbo;
    mll128 ha3=ha1*Bn2r + Brep;
    return Hash{ha3,n1+n2*r};
  };

  auto calchash=[&](vector<pair<string,ll>> &st){
    Hash h{0,0};
    for(auto&&[s,t]:st){
      Hash h2=calcstringhash(s);
      //Hash h3=hashrepeat(h2,t);
      //h=hashconcat(h,h3);
      h=repeatconcat(h,h2,t);
    }
    return h;
  };

  auto ha=calchash(ST);
  ST.reverse();
  for(auto&&[s,t]:ST) reverse(all(s));
  auto revha=calchash(ST);
  if(ha==revha){
    cout << "Yes" << '\n';
  }
  else{
    cout << "No" << '\n';
  }
  return;
}

}//SorvingSpace

//////////////////////////////////////////

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;
}