#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 ----*/
  mll_(){}
  mll_(ll v): val_(v){ norm(); }
  mll_(ll v,bool b): val_(v){} //正規化無のコンストラクタ
  Int val()const{ return (Int)val_; }
  bool isnone() const { return val_==-1; } //true:値なし
  mll_ &none() { val_=-1; return *this; } //値なしにする
  mll_ &inv(){ val_=modinv((int)val_); return *this; }
  mll_ &operator+=(mll_ b){ val_+=b.val_; return normP(); }
  mll_ &operator-=(mll_ b){ val_-=b.val_; return normS(); }
  mll_ &operator*=(mll_ b){ val_*=b.val_; return normR(); }
  mll_ &operator/=(mll_ b){ return *this*=b.inv(); }
  mll_ &operator+=(ll b){ return *this+=mll_(b); }
  mll_ &operator-=(ll b){ return *this-=mll_(b); }
  mll_ &operator*=(ll b){ return *this*=mll_(b); }
  mll_ &operator/=(ll b){ return *this/=mll_(b); }
  mll_ operator-()const{ return mll_(*this).invsg(); }
  mll_ operator+(mll_ b)const{ return mll_(*this)+=b; }
  mll_ operator-(mll_ b)const{ return mll_(*this)-=b; }
  mll_ operator*(mll_ b)const{ return mll_(*this)*=b; }
  mll_ operator/(mll_ b)const{ return mll_(*this)/=b; }
  mll_ operator+(ll b)const{ return mll_(*this)+=b; }
  mll_ operator-(ll b)const{ return mll_(*this)-=b; }
  mll_ operator*(ll b)const{ return mll_(*this)*=b; }
  mll_ operator/(ll b)const{ return mll_(*this)/=b; }
  friend mll_ operator+(ll a,mll_ b){ return b+a; }
  friend mll_ operator-(ll a,mll_ b){ return -b+a; }
  friend mll_ operator*(ll a,mll_ b){ return b*a; }
  friend mll_ operator/(ll a,mll_ b){ return mll_(a)/b; }
  bool operator==(mll_ b)const{ return val_==b.val_; }
  bool operator!=(mll_ b)const{ return val_!=b.val_; }
  bool operator==(ll b)const{ return *this==mll_(b); }
  bool operator!=(ll b)const{ return *this!=mll_(b); }
  friend bool operator==(ll a,mll_ b){ return mll_(a)==b; }
  friend bool operator!=(ll a,mll_ b){ return mll_(a)!=b; }
  friend ostream &operator<<(ostream &os,mll_  a){ return os << a.val_; }
  friend istream &operator>>(istream &is,mll_ &a){ return is >> a.val_; }
  mll_ pow(ll k)const{
    mll_ ret(1,false),a(*this);
    for (; k>0; k>>=1,a*=a) if (k&1)ret*=a;
    return ret;
  }
  static constexpr int mod() { return MOD; }
  //enum{ modll=MOD };
};


struct bll{ //第0～62bitまで対応
  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; }
  bool contains(ll b)const{ return (s&b)==b; }
  bll &oneq  (ll l,ll r){ s|= rangemask(l,r); return *this; }//l>=r
  bll &offeq (ll l,ll r){ s&=~rangemask(l,r); return *this; }//l>=r
  bll &flipeq(ll l,ll r){ s^= rangemask(l,r); return *this; }//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 l,ll r)const{ return bll(s).oneq  (l,r); }//l>=r
  bll off (ll l,ll r)const{ return bll(s).offeq (l,r); }//l>=r
  bll flip(ll l,ll r)const{ return bll(s).flipeq(l,r); }//l>=r
  bll on  (ll i)const{ return bll(s).oneq  (i); }
  bll off (ll i)const{ return bll(s).offeq (i); }
  bll flip(ll i)const{ return bll(s).flipeq(i); }
  bll substr(ll l,ll r)const{ return (s&rangemask(l,r))>>r; }//l>=r
  static ll rangemask(ll l,ll r){ return (1LL<<(l+1))-(1LL<<r); }//l>=r
  ll msbit()const{
    for(ll o=63,x=-1;;){
      ll m=(o+x)/2;
      if(s < (1LL<<m)) o=m; else x=m;
      if(o-x==1) return x;
    }
  }
  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(); }
  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()const{
    string ret;
    for(ll i=msbit();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
struct args{
  using Int = long long;
  args(){}
  args &wd(Int wd__){ (void)wd__; 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::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作成
  template<class S> Vector(const Vector<S>& b):vT(b.begin(),b.end()){}
  template<class S> Vector(const vector<S>& b):vT(b.begin(),b.end()){}
  Vector(Int n,T s,T d){ iota(n,s,d); }
  Vector(Int n,function<T(Int)> g):vT(n){ for(Int i=0;i<n;++i) (*this)[i]=g(i); }
  template<class S> Vector &operator+=(S x){ for(T &e: *this) e+=x;  return *this; }
  template<class S> Vector &operator-=(S x){ for(T &e: *this) e-=x;  return *this; }
  template<class S> Vector &operator*=(S x){ for(T &e: *this) e*=x;  return *this; }
  template<class S> Vector &operator/=(S x){ for(T &e: *this) e/=x;  return *this; }
  template<class S> Vector &operator%=(S x){ for(T &e: *this) e%=x;  return *this; }
  template<class S> Vector operator+(S x)const{ return Vector(*this)+=x; }
  template<class S> Vector operator-(S x)const{ return Vector(*this)-=x; }
  template<class S> Vector operator*(S x)const{ return Vector(*this)*=x; }
  template<class S> Vector operator/(S x)const{ return Vector(*this)/=x; }
  template<class S> Vector operator%(S x)const{ return Vector(*this)%=x; }
  Vector &operator--(int){ return *this-=T(1); }
  Vector &operator++(int){ return *this+=T(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; }
  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.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)
*/

#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<class E> struct HeavyLightDecomposition{
  using Int = long long;
  vector<vector<E>> *to=nullptr; //グラフ
  vector<Int> head; //head[v] 頂点vの属する重パスの根側端頂点
  vector<Int> id;   //id[v] 頂点vの行きがけ順番号 0-origin
  vector<Int> rev;  //rev[ID] 行きがけ順番号IDの頂点v
  vector<Int> lst;  //lst[v] 頂点vのサブ木内頂点の行きがけ順番号最大値
  vector<Int> par;  //par[v] 頂点vの親頂点  par[根]は-1
  vector<Int> siz;  //siz[v] 頂点vのサブ木内頂点数(v含む)
  vector<Int> dpt;  //dpt[v] 頂点vの深さ(根が0)
  HeavyLightDecomposition(){}
  HeavyLightDecomposition(vector<vector<E>> &to_,Int root=0):
    to(&to_),head(size(),root),id(size()),rev(size()),lst(size()),
    par(size()),siz(size(),1),dpt(size())
  {
    assert(!to_.empty());
    dfsSz(root);
    dfsHld(root);
  }
  Int size()const{ return to->size(); } //頂点数
  Int vof(Int id_)const{ return rev[id_]; }
  Int idof(Int v)const{ return id[v]; }
  Int parentof(Int v)const{ return par[v]; }
  Int subszof(Int v)const{ return siz[v]; }
  Int depthof(Int v)const{ return dpt[v]; }
  bool isleaf(Int v)const{
    return id[v]==size()-1 || head[v]!=head[rev[id[v]+1]];
  }
  vector<pair<Int,Int>> preorder()const{ //行きがけ順の(頂点,親頂点)列
    vector<pair<Int,Int>> ret;
    for(int i=0;i<size();++i) ret.emplace_back(rev[i],par[rev[i]]);
    return ret;
  }
  Int lcaorg(Int u,Int v)const{ //u,vのLCA
    while(head[u]!=head[v]){
      if(id[u]>id[v]) swap(u,v);
      v=par[head[v]];
    }
    return id[u]<id[v] ? u : v;
  }
  Int lca(Int u,Int v)const{ //u,vのLCA
    if(id[u]>id[v])swap(u,v);
    while(head[u]!=head[v] && id[u]<id[v]) v=par[head[v]];
    return id[u]<id[v] ? u : v;
  }

  Int dist(Int u,Int v)const{ return dpt[u]+dpt[v]-2*dpt[lca(u,v)]; }
  Int jump(Int v,Int k)const{ //vのk個上 なければ-1
    while(id[v]-k<id[head[v]]){
      k-=id[v]-id[head[v]]+1;
      v=par[head[v]];
      if(v==-1) return -1;
    }
    return rev[id[v]-k];
  }
  vector<pair<Int,Int>> heavypaths()const{ //[stID0,enID0],[stID1,enID1],…
    vector<pair<Int,Int>> ret;
    Int enID=to->size()-1;
    while(enID!=-1){
      Int stID=id[head[rev[enID]]];
      ret.emplace_back(stID,enID);
      enID=stID-1;
    }
    reverse(ret.begin(),ret.end());
    return ret;
  }
  vector<pair<Int,Int>> uv2idranges(Int u,Int v,bool isedge=false)const
  {                                                //パスu→vを重パス分解
    Int a=lca(u,v);
    //aを含む重パス上まで移動
    auto func=[&](Int &t){
      vector<pair<Int,Int>> t2a;
      while(head[t]!=head[a]){
        t2a.emplace_back(id[t],id[head[t]]);
        t=par[head[t]];
      }
      return t2a;
    };
    vector<pair<Int,Int>> u2a=func(u),v2a=func(v);

    //aを含む重パス上の追加
    if(!isedge){//頂点
      if(u!=a)      u2a.emplace_back(id[u],id[a]);
      else if(v!=a) v2a.emplace_back(id[v],id[a]);
      else           u2a.emplace_back(id[a],id[a]);
    }
    else{//辺
      if(u!=a)      u2a.emplace_back(id[u],id[a]-1);
      else if(v!=a) v2a.emplace_back(id[v],id[a]-1);
    }
    //v2aを反転して結合
    while(!v2a.empty()){
      auto [st,en]=v2a.back();  v2a.pop_back();
      u2a.emplace_back(en,st);
    }
    return u2a;
  }
  vector<Int> path(Int u,Int v)const{ //パスu→v 頂点列
    Int a=lca(u,v);
    auto func=[&](Int t){
      vector<Int> t2a;
      for(;t!=a;t=par[t]) t2a.push_back(t);
      return t2a;
    };
    vector<Int> u2a=func(u),v2a=func(v);
    u2a.push_back(a);
    while(!v2a.empty()){
      u2a.push_back(v2a.back());  v2a.pop_back();
    }
    return u2a;
  }
  pair<Int,Int> subtree2idrange(Int v)const{ //サブ木ID範囲
    return {id[v],lst[v]};
  }
private:
  void dfsSz(Int v,Int p=-1){
    par[v]=p;
    for(auto&& u: (*to)[v]){
      if(u==p)continue;
      dpt[u]=dpt[v]+1;
      dfsSz(u,v);
      siz[v]+=siz[u];
    }
  }
  Int idcnt=0;
  void dfsHld(Int v,Int p=-1){
    vector<E> &tov=(*to)[v];
    id[v]=idcnt++;
    rev[id[v]]=v;
    //重パスの子を先頭へ
    Int mxsz=-1;
    for(auto&& u: tov){
      if(u==p)continue;
      if(mxsz<siz[u]) mxsz=siz[u],swap(tov[0],u);
    }
    //head決めるdfs
    for(auto&& u: tov){
      if(u==p)continue;
      head[u] = (u==(*to)[v][0]) ? head[v] : u;
      dfsHld(u,v);
    }
    lst[v]=idcnt-1;
  }
  Int root()const{ return rev[0]; }
};
/*
- 定義                      ↓木グラフ to[v][0]が重パスになるよう変更される
HeavyLightDecomposition hld(to);
HeavyLightDecomposition hld(to,root);
- 行きがけ順番号id ⇔ 頂点v
ll v=hld.vof(id);
ll id=hld.idof(v);
- 親頂点
ll p=hld.parentof(v);
- サブ木内頂点数(v含む)
ll s=hld.subszof(v);
- 深さ(根が0)
ll d=hld.depthof(v);
- 葉?
bool b=hld.isleaf(v);
- 行きがけ順の頂点列
vpll vp=hld.preorder();
for (auto&&[v,p]:vp){
  for (ll u:to[v]){
    if (u==p)continue;
    //
  }
}
- LCA
ll a=hld.lca(u,v);
- k個上の親頂点  なければ-1
ll u=hld.jump(v,k);
- 距離
ll d=hld.dist(u,v);
- ID列上の重パス列挙 [stID0,enID0],[stID1,enID1],…
vpll hs=hld.heavypaths();
- パスu→vを重パス分解  [uID,enID0],[stID1,enID1],…,[stID*,vID]
.                       u→vの順のためstID>enIDもあり
vpll hs=hld.uv2idranges(u,v);
vpll hs=hld.uv2idranges(u,v,true); //辺の値を乗せる時(LCA含まない)
- パスu→vの頂点列
vll path=hld.path(u,v);
- ID列上のサブ木範囲
auto[l,r]=hld.subtree2idrange(v);
*/

struct binarytrie{
  struct node{
    ll nm;
    node *c[2];
    //vll accept;
    node():nm(0),c{nullptr,nullptr} {}
  };
  ll B=59;//ビット数-1(第Bビットが最上位bit)
  node *root=nullptr;
  binarytrie(){}
  binarytrie(ll B_){ init(B_); }
  void init(ll B_){ B=B_-1; root=NewNode(); }
  ll size(){ return root->nm; }//要素数
  bool empty(){ return root->nm==0; }
  void insert(ll x,ll delta=1,ll idx=-1){ assert(IsIn(x)); Add(root,x,idx,B,delta); }
  void erase(ll x){ insert(x,-1); }
  bool contains(ll x){ return (*this)[x]>0; }
  ll rank(ll x,ll xorval=0){return leftnumof(x-1,xorval)+1;}  //xの順位(昇順) 1-origin
  ll rrank(ll x,ll xorval=0){return rightnumof(x+1,xorval)+1;}//xの順位(降順) 1-origin
  ll leftnumof(ll x,ll xorval=0){ //x以下の個数
    if (x<0) return 0;
    if (x+1 >= 1LL<<B<<1) return size();
    return CountLess(root,x+1,B,xorval);
  }
  ll rightnumof(ll x,ll xorval=0){ //x以上の個数
    if (x<=0) return size();
    if (x >= 1LL<<B<<1) return 0;
    return size()-CountLess(root,x,B,xorval); }
  ll operator[](ll x) { node *v=find(x); return v?v->nm:0; }//xの個数
  ll min(ll xorval=0){ assert(!empty()); return nthsmall(1,xorval); }//最小値
  ll max(ll xorval=0){ assert(!empty()); return nthbig(1,xorval); }//最大値
  ll nthsmall(ll r,ll xorval=0){return nthnode(r,xorval).first;}//r番目(昇順) 1-origin
  ll nthbig(ll r,ll xorval=0){return nthnode(size()+1-r,xorval).first;}//r番目(降順) 1-origin
  node *find(ll x){ return IsIn(x) ? Find(root,x,B) : nullptr; }//なければnullptr
  pair<ll,node*> nthnode(ll r,ll xorval=0){//r番目(昇順)の要素とnode 1-origin
    if (r<=0)return{-INF,nullptr};
    if (r>root->nm)return{INF,nullptr};
    return NthNode(root,r-1,B,xorval);
  }
  void Add(node *v,ll x,ll idx,ll depth,ll delta){
    if (depth==-1){
      //if (idx>=0)t->accept.push_back(idx);
    }
    else{
      node *&c = v->c[(x>>depth)&1];
      if (!c) c=NewNode();
      Add(c,x,idx,depth-1,delta);
    }
    v->nm+=delta;   assert(v->nm>=0);
  }
  node *Find(node *v,ll x,ll depth){
    if (depth==-1) return v;
    node *c = v->c[(x>>depth)&1];
    return c ? Find(c,x,depth-1) : nullptr;
  }
  pair<ll,node*> NthNode(node *v,ll r,ll depth,ll xorval){//サブ木v内r番目の要素 0-origin
    if (depth==-1) return{0,v};
    bool f=(xorval>>depth)&1;
    ll cnm = v->c[f] ? v->c[f]->nm : 0;
    if (cnm<=r){
      auto ret=NthNode(v->c[f^1],r-cnm,depth-1,xorval);
      ret.first|=1LL<<depth;
      return ret;
    }
    else return NthNode(v->c[f],r,depth-1,xorval);
  }
  ll CountLess(node *v,ll x,ll depth,ll xorval){//サブ木v内x未満個数
    if (depth==-1)return 0;
    ll ret=0;
    bool f=(xorval>>depth)&1, g=(x>>depth)&1;
    if (g && v->c[f]) ret+=v->c[f]->nm;
    if (v->c[f^g]) ret+=CountLess(v->c[f^g],x,depth-1,xorval);
    return ret;
  }
  bool IsIn(ll x){ return 0<=x && x<(1LL<<B<<1); }
  #if defined(MYLOCAL)//メモリ管理有
  vector<node*> buf;
  node *NewNode(){ buf.push_back(new node()); return buf.back(); }
  ~binarytrie(){ for (node *p:buf) delete p; }
  #else//メモリ管理無
  node *NewNode(){ return new node(); }
  #endif
  void dump(){ for(int r=1;r<=size();r++)cerr<<nthsmall(r)<<' '; cerr<<'\n'; }
};
/*
binarytrie bt(n); //0≦x＜2^nならnをセット
ll s=bt.size();
bool b=bt.empty();
bt.insert(x); //xを1つ挿入
bt.insert(x,nm); //xをnm個挿入
bt.erase(x); //xを1つ削除
bool b=bt.contains(x); //x∈集合ならtrue
ll r=bt.rank(x);         //xの順位(昇順) 1-origin
ll r=bt.rank(x,xorval);  //xの順位(昇順) 1-origin (集合にxorvalをかけた時)
ll r=bt.rrank(x);        //xの順位(降順) 1-origin
ll r=bt.rrank(x,xorval); //xの順位(降順) 1-origin (集合にxorvalをかけた時)
ll nm=bt.leftnumof(x);         //x以下の個数
ll nm=bt.leftnumof(x,xorval);  //x以下の個数 (集合にxorvalをかけた時)
ll nm=bt.rightnumof(x);        //x以上の個数
ll nm=bt.rightnumof(x,xorval); //x以上の個数 (集合にxorvalをかけた時)
ll nm=bt[x]; //xの個数
ll x=bt.min();       //最小値
ll x=bt.min(xorval); //最小値 (集合にxorvalをかけた時の最小値。xorがかかったものを返す)
ll x=bt.max();       //最大値
ll x=bt.max(xorval); //最大値 (集合にxorvalをかけた時の最大値。xorがかかったものを返す)
ll x=bt.nthsmall(r);        //r番目の要素(昇順) 1-origin ない時はINF
ll x=bt.nthsmall(r,xorval); //              〃                (集合にxorvalをかけた時)
ll x=bt.nthbig(r);          //r番目の要素(降順) 1-origin ない時は-INF
ll x=bt.nthbig(r,xorval);   //              〃                (集合にxorvalをかけた時)
*/


void cin2solve()
{
  auto N=cin1<ll>();
  auto uv=cinv<pair<ll,ll>>(N-1);
  for(auto&&[u,v]:uv) u--,v--;
  auto S_=cin1<string>();

  vll S=str2num(S_,'0',"");
  S*=2; S--;

  auto makeGraph=[](ll n,vector<pair<ll,ll>> &ab,bool isDirected){
    vvll to(n);
    for(auto [a,b]:ab){
      to[a].push_back(b);
      if(!isDirected) to[b].push_back(a);
    }
    return to;
  };
  vvll to=makeGraph(N,uv,false);
  HeavyLightDecomposition hld(to);

  if(N==1){
    cout << S[0] << '\n'; return;
  }

  Vector dp(N,binarytrie());
  vll bases(N,N);//0の入る位置 dp[v]には+baseしてから入れる
  for(auto&& b: dp){
    b.init(bll(2*N+10).msbit()+1);
  }
  auto add=[&](binarytrie &b,ll base,ll va,ll nm){
    b.insert(va+base,nm);
  };
  auto min_pop=[&](binarytrie &b,ll base)->pll{
    ll x=b.min();
    ll nx=b[x];
    b.insert(x,-nx);
    return {x-base,nx};
  };
  auto overnmget=[&](binarytrie &b,ll base,ll x){
    return b.rightnumof(x+base);
  };


  ll ans=0;
  vpll vp=hld.preorder().reverse();
  for (auto&&[v,p]:vp){
    if(hld.isleaf(v)){
      add(dp[v],bases[v],S[v],1);
      if(S[v]==1) ans++;
    }
    else{
      vll us;
      for(ll u:to[v]){
        if(u==p)continue;
        us.push_back(u);
      }
      ll U=sz(us);
      us.sort([&](ll u,ll uu){return hld.subszof(u)>hld.subszof(uu);});
      //vが片端のパスをカウント
      if(S[v]==1) ans++;
      for(auto&& u: us){
        binarytrie &b =dp[u];
        ll base=bases[u];
        ll overnm=overnmget(b,base,-S[v]+1);
        ans+=overnm;
      }
      //us[0]に集約しながら、異なる子をまたぐパスをカウント
      rep(i,1,U-1){
        ll u=us[0];
        ll uu=us[i];
        binarytrie &b =dp[u];
        binarytrie &bb=dp[uu];
        ll base =bases[u];
        ll bbase=bases[uu];
        vpll bf;
        while(sz(bb)>0){
          auto [x,nm]=min_pop(bb,bbase);
          bf.emplace_back(x,nm);
        }
        //まずカウント
        for(auto&&[x,nm]:bf){
          ll va=x+S[v];
          ll overnm=overnmget(b,base,-va+1);
          ans+=nm*overnm;
        }
        //集約
        for(auto&&[x,nm]:bf) add(b,base,x,nm);
      }
      //vへmove,vを追加
      {
        ll u=us[0];
        dp[v]=move(dp[u]);
        bases[v]=bases[u]-S[v];
        add(dp[v],bases[v],S[v],1);
      }
    }
  }
  cout << ans << '\n';

  return;
}

};//SolvingSpace

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

int main(){
  #if 1
  //SolvingSpace::labo();
  SolvingSpace::cin2solve();
  //SolvingSpace::generand();
  #else
  ll t;  cin >> t;
  rep(i,0,t-1){
    SolvingSpace::cin2solve();
    //SolvingSpace::generand();
  } 
  #endif
  cerr << timeget() <<"ms"<< '\n';
  return 0;
}