#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},};



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); }); }


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 ? 1 : -1; }
};


template<class STATE,class VAL,class Ope,class Tran,class ToID,class FromID>
struct PushBasedDP{
  using Int = long long;
  using ll = long long;
  Ope ope; //配るときの演算
  Tran tran;
  ToID toID;
  FromID fromID;
  VAL ini,dmy; //ini:dpテーブルの初期値
  const bool isinf; //true:iniは配らない(ini=±INFを想定)
  vector<VAL> dp_;
  bool autoResize = true;
  bool nextdp = false; //true:nextDP
  //-------- コンストラクタ --------
  PushBasedDP(STATE,VAL,Ope ope_,Tran tran_,ToID toID_,FromID fromID_,
    VAL ini_,bool isinf_)
    : ope(ope_),tran(tran_),toID(toID_),fromID(fromID_),
    ini(ini_),dmy(ini_),isinf(isinf_){}
  //事前メモリ確保用コンストラクタ
  template<class RNG> PushBasedDP(STATE,VAL,Ope ope_,Tran tran_,ToID toID_,
    FromID fromID_,VAL ini_,bool isinf_,const RNG &rng)
    : ope(ope_),tran(tran_),toID(toID_),fromID(fromID_),
    ini(ini_),dmy(ini_),isinf(isinf_),
    dp_(rng.isCycle() ? rng.csize()*rng.cycle() : rng.size(),ini),
    autoResize(false),nextdp(rng.isCycle()){}
  //-------- アクセス・計算実行 --------
  template<class... Args> VAL &operator()(Args&&... args){ return Dp(STATE{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,ini);  return vVAL(dp_);
  }
  vvVAL tov(Int xl,Int yl){
    dp_.resize(xl*yl,ini);
    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,ini);
    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,ini);
    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){
    ll id=toID(vx);
    if(autoResize && (ll)dp_.size()<=id) dp_.resize(id+1,ini);
    return id<0 ? (dmy=ini) : dp_[id];
  }
  void calcSt(const STATE &vx){
    auto dstVal=[&](const STATE &nvx,VAL val){
      VAL &DpNvx=Dp(nvx);
      VAL nval=ope(DpNvx,val);
      if(DpNvx!=nval) DpNvx=nval,preRenew(nvx,vx);
    };
    VAL va=Dp(vx);
    if(!(isinf && va==ini)) tran(vx,va,dstVal);
  }
  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){
    auto [sx,ex,xl,dx]=rng.SELDx();
    for(ll xx=0;xx<xl;++xx){
      //1つ前を消去
      if(xx>0){
        ll id=toID(rng.start(xx-1));
        auto it=dp_.begin()+id;
        fill(it,it+rng.csize(),ini);//dp_[id～id+csize-1]を消去
      }
      //現在位置xxを処理
      STATE vx=rng.start(xx);
      do calcSt(vx); while(rng.next(vx,1));
    }
  }
  //-------- ダンプ --------
public:
#if 1
  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).tostd(),args().rngs(rng.rngs()).labels(labels).tr(axes));
  }
#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)const{ (void)vx,(void)src; }
#endif
};



auto solve(
  ll N,ll M,vpll cd_
){
  cd_.rsort();
  wrapv cd=cd_;
  using STATE=array<ll,2>;
  using VAL=ll;
  auto operation=[&](VAL a,VAL b){return max(a,b);};
  VAL ini=-INF;  bool isinf=true;

  auto transition=[&](const STATE &vx,VAL va,auto dstVal){
    auto[i,m]=vx;
    auto [c,d]=cd[i+1];
    dstVal(STATE{i+1,m}  ,va);
    if (m+c<=M) dstVal(STATE{i+1,M-(M-(m+c))/2},va+d);
  };

  auto [s1,e1]=pll{-1,N-1}; //1つ目の変数の範囲
  auto [s2,e2]=pll{0,M};    //2つ目の変数の範囲
  rangeNd<2> rng({s1,s2},{e1,e2},2);
  PushBasedDP dp(STATE(),VAL(),operation,transition,rng.toID(),rng.fromID(),ini,isinf,rng);

  dp(-1,0)=0;
  dp.calcall(rng);
  ll ans=0;
  rep(m,0,M){
    chmax(ans,dp(N-1,m));
  }
  //dp.dump(rng,{"i","m"},{0,1});


  return ans;
}


struct permutation_enumerate{
  ll N; vll p;
  permutation_enumerate(){}
  permutation_enumerate(ll n){ init(n); }
  void init(ll n){ N=n; p.resize(N); iota(p.begin(),p.end(),0); }
  bool next(){ return next_permutation(p.begin(),p.end()); }
};

auto solve2(
  ll N,ll M,vpll cd
){
  ll ans=0;
  permutation_enumerate pe(N);
  for (bool is=true; is; is=pe.next()){
    ll csm=0;
    ll dsm=0;
    rep(I,0,N-1){
      ll i=pe.p[I];
      auto [c,d]=cd[i];
      csm+=c*Pow(2,I);
      if(csm>M) break;
      dsm+=d;
    }
    chmax(ans,dsm);
  }

  return ans;
}


void solvecomp(
  ll N,ll M,vpll cd
){
  auto coutans=[](auto ans){
    /*
    出力を書く
    */
    cout << ans << '\n';
  };

  auto ans=solve(
    N,M,cd
  );
  coutans(ans);
#if 0
  cout << "- - - - -\n";
  auto an2=solve2(
    N,M,cd
  );
  coutans(an2);
  cout << "================\n";
  if(ans!=an2){
    cout << "============ input =============\n";
    /*
    inputファイルの形式で吐き出す
    */
    cout << N << ' ' << M << '\n';
    for(auto [c,d]:cd){
      cout << c << ' ' << d << '\n';
    }
    cout << "============ input end =========\n\n================\n";
    getchar();  rep(i,0,10000000000) i++;//getchar()の代わり
  }
#endif
}

void cin2solve(){
  /*
  cinからの入力を書く
  */
  auto [N,M]=cins<ll,ll>();
  auto cd=cinv<pair<ll,ll>>(N);

  solvecomp(
    N,M,cd
  );
}




struct rndutil{
  using Int = long long;
  mt19937 mt;
  rndutil(Int seed=0):mt((unsigned int)seed){}
  Int val(Int a,Int b){ return mt()%(b-a+1)+a; }//[a,b]の乱数
  dd dval(){ return (dd)mt()/(1ll<<32); }//[0,1)の浮動小数点乱数
  vector<Int> ary(Int n,Int a,Int b){//長さnの配列、要素[a,b]内、重複可
    vector<Int> v(n);
    for(Int i=0;i<n;++i) v[i]=this->val(a,b);
    return v;
  }
  vdd dary(Int n){//長さnの[0,1)の浮動小数点乱数
    vdd v(n);
    for(Int i=0;i<n;++i) v[i]=this->dval();
    return v;
  }
  string str(Int n,char cs,char ce){//長さnのstring、使用文字cs~ce、重複可
    string str(n,cs);
    for(Int i=0;i<n;++i) str[i]+=(char)this->val(0,ce-cs);
    return str;
  }
  vector<Int> sample(Int n,Int a,Int b){//長さnの配列、要素[a,b]内、重複無
    Int len=b-a+1;
    assert(n<=len);
    vector<Int> v(len);
    iota(v.begin(),v.end(),a);
    for(Int i=0;i<n;++i) swap(v[i],v[this->val(i,len-1)]);
    v.resize(n);
    return v;
  }
  template <class T> auto sample(const T &v){//1つ選択
    return v[val(0,(Int)v.size()-1)];
  }
  template <class T> auto samplepop(T &v){//1つ取り出し(末尾swapし削除)
    Int i=val(0,(Int)v.size()-1);
    auto ret=v[i];
    v[i]=v.back();
    v.pop_back();
    return ret;
  }
  Int weightedSampleCore(const vector<Int> &cm){//重み付き乱択 累積和版
    Int randval=val(1,cm.back());
    return Int(lower_bound(cm.begin(),cm.end(),randval)-cm.begin());
  }
  Int weightedSample(const vector<Int> &pr){//重み付き乱択
    vector<Int> cm=pr;
    for(Int i=1;i<(Int)cm.size();++i) cm[i]+=cm[i-1];
    return weightedSampleCore(cm);
  }
  template <class T> void shuffle(T &v){//配列や文字列をシャッフル
    Int n=(Int)v.size();
    for(Int i=0;i<n;++i) swap(v[i],v[this->val(i,n-1)]);
  }
  pair<Int,Int> range(Int a,Int b,Int m=1){ //[a,b]内閉区間長さm以上
    while(true){
      Int l=this->val(a,b+1),r=this->val(a,b+1);
      if(r-l<m)continue;
      return {l,r-1};
    }
  }
  pair<Int,vector<pair<Int,Int>>> graph(Int n){ //単純連結無向グラフ 木に近いものが多め
    while(true){
      vector<pair<Int,Int>> uv;
      UnionFindTree uft(n);
      for(Int v=0;v<n;++v) for(Int u=v+1;u<n;++u){
        if(this->val(1,n*(n-1)/2)>=n)continue;//各辺確率n/Mで採用(M:総辺数n(n-1)/2)
        uv.emplace_back(v,u);
        uft.Unite(v,u);
      }
      if(uft.grpnm()==1) return {(Int)uv.size(),uv};
    }
  }
  vector<pair<Int,Int>> tree(Int n){ //木
    vector<pair<Int,Int>> uv;
    UnionFindTree uft(n);
    while(uft.grpnm()>1){
      Int u=this->val(0,n-1),v=this->val(0,n-1);
      if(uft.Same(u,v)) continue;
      if(u>v)swap(u,v);
      uft.Unite(u,v);
      uv.emplace_back(u,v);
    }
    return uv;
  }
  struct UnionFindTree{
    vector<Int> par,rank,size;
    Int g;
    UnionFindTree(Int num): par(num),rank(num),size(num,1),g(num){
      for(Int i=0;i<num;++i) par[i] = i;
    }
    bool Same(Int x,Int y) { return Root(x) == Root(y); }
    Int Root(Int x) { return (par[x] == x) ? x : par[x]=Root(par[x]); }
    Int Size(Int x){ return size[Root(x)]; }
    Int grpnm() { return g; }
    void Unite(Int x,Int y){
      x = Root(x);
      y = Root(y);
      if(x == y) return;
      if(rank[x] < rank[y]) par[x] = y;
      else                   par[y] = x;
      if(rank[x] == rank[y]) rank[x]++;
      size[x] = size[y] = size[x] + size[y];
      g--;
    }
  };
};
/*
rndutil ru;
- -------- 整数x -------- 区間[a,b]
ll x=ru.val(a,b);
- -------- 浮動小数点数x -------- [0,1)
dd x=ru.dval();
- -------- 整数ベクトルv -------- 長さn,要素の区間[a,b] 配列重複有
vll v=ru.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=ru.sample(n,a,b);
- -------- 浮動小数点ベクトルv -------- 長さn,[0,1)
vdd v=ru.dary(n);
- -------- 文字列s -------- 長さn,使用文字'a'~'z'
string s=ru.str(n,'a','z');
- -------- 乱択 -------- vll,string等から1つ選択
ll   x=ru.sample(v);
char c=ru.sample(s);
- -------- 乱択削除 -------- vll,string等から1つ取り出し(末尾swapし削除)
ll x=ru.samplepop(v);
- -------- 重み付き乱択 -------- i∈{0,1,…,n-1}を重みwiで選択
ll i=ru.weightedSample(w);
ll i=ru.weightedSampleCore(cm);
.                          ↑wの累積和を与えることで高速に
- -------- シャッフル -------- vll,string等
ru.shuffle(a);
- -------- 範囲[l,r] --------  [l,r]⊆[a,b] 区間長1以上
auto[l,r]=ru.range(a,b);
auto[l,r]=ru.range(a,b,m);
.                      ↑区間長m以上
- -------- 木(辺集合uv) --------  頂点数n
vpll uv=ru.tree(n);
- -------- 単純連結無向グラフ(辺集合uv,辺数m) --------  頂点数n
auto[m,uv]=ru.graph(n);
*/


void generand(){
  rndutil ru;

  rep(q,0,INF){
    /*
    乱数による入力の生成を書く
    */
    ll N=ru.val(1,3);
    ll M=ru.val(0,10);
    vpll cd;
    rep(i,0,N-1){
      ll c=ru.val(0,5);
      ll d=ru.val(1,5);
      cd.emplace_back(c,d);
    }

    solvecomp(
      N,M,cd
    );
  }
}

}//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;
}