#include <bits/stdc++.h>
#define REP(i,n) for(int i=0,i##_len=(n);i<i##_len;++i)
#define RREP(i,n) for(int i=int(n)-1;i>=0;--i)
#define rep(i,a,b) for(int i=int(a);i<int(b);++i)
#define rrep(i,a,b) for(int i=int(a)-1;i>=int(b);--i)
#define All(x) (x).begin(),(x).end()
#define rAll(x) (x).rbegin(),(x).rend()
#define ITR(i,x) for(auto i=(x).begin();i!=(x).end();++i)
using namespace std;
using Graph = vector<vector<int>>;

typedef long long ll;
typedef pair<ll, ll> P;
typedef vector<int> vi;
typedef vector<vi> vvi;
typedef vector<ll> vl;
typedef vector<vl> vvl;
constexpr ll mod = 1e9+7;
constexpr double eps = 1e-9;
const double PI = acos(-1);
void solve();
P bisearch(ll l,ll r,function<bool(ll)> f){
    while((l+1)<r){
        ll m=(l+r)/2;
        if(f(m)) r=m;
        else l=m;
    }
    return P(l,r);
}
ll SQRT(ll n){if(n==1) return 1;return bisearch(0,n,[n](ll x){return x>n/x;}).first;}
ll roundup(ll n,ll div){
    if(n>0) return (n-1)/div+1;
    else return n/div;    
}
bool square(ll a){ll n=SQRT(a);return a==n*n;}
ll npow(ll x, ll n){
    ll ans = 1;
    while(n != 0){
        if(n&1) ans = ans*x;
        x = x*x;
        n = n >> 1;
    }
    return ans;
}
ll mpow(ll x, ll n){ //x^n(mod) ←普通にpow(x,n)では溢れてしまうため，随時mod計算 2分累乗法だから早い
    ll ans = 1;
    while(n != 0){
        if(n&1) ans = ans*x % mod;
        x = x*x % mod;
        n = n >> 1;
    }
    return ans;
}
ll inv_mod(ll a){return mpow(a,mod-2);}
int digitsum(ll N,int a){
    if(N==0) return 0;
    int ret=0;
    ret+=digitsum(N/a,a)+N%a;
    return ret;
}
ll gcd(ll x,ll y){return y ? gcd(y,x%y) : x;};//xとyの最大公約数
ll lcm(ll x,ll y){return x/gcd(x,y)*y;}//xとyの最小公倍数
void YN(bool flg){std::cout<<(flg?"YES":"NO")<<endl;}
void Yn(bool flg){std::cout<<(flg?"Yes":"No")<<endl;}
void yn(bool flg){std::cout<<(flg?"yes":"no")<<endl;}
P splitint(string n,int a){
    int Len=n.length();
    if(a<0||Len<a) return {-1,-1};
    string left,right;
    if(a!=0) left=n.substr(0,a);
    if(a!=Len) right=n.substr(a);
    return P(stoll(left),stoll(right));
}
ll manhattan(const P &a,const P &b){return llabs(a.first-b.first)+llabs(a.second-b.second);}
bool kaibun(string s){//O(|S|)
    string  sdash=s;
    reverse(All(s));
    return s==sdash;
}
class Ruiseki{
  private:
    vector<ll> LEFT,RIGHT;
    function<ll(ll,ll)> F;
    int N;
  public:
    Ruiseki(ll INI,const vector<ll> &a,function<ll(ll,ll)> f){
        N=a.size();F=f;
        LEFT.resize(N+1);RIGHT.resize(N+1);
        LEFT[0]=RIGHT[0]=INI;
        REP(i,N){ 
            LEFT[i+1]=F(LEFT[i],a[i]);
            RIGHT[i+1]=F(RIGHT[i],a[N-i-1]);
        }
    }
    ll out(int l,int r){//[l,r)の外の累積
        return F(LEFT[l],RIGHT[N-r]);
    }
};
class mint {
 private:
  ll _num,_mod;
  mint set(ll num){ 
      _num = num ;
      if(_num>=0) _num%=_mod;
      else _num+=(1-(_num+1)/_mod)*_mod; 
      return *this;
  }
  ll _mpow(ll x, ll n){ //x^n(mod) ←普通にpow(x,n)では溢れてしまうため，随時mod計算 2分累乗法だから早い
    ll ans = 1;
    while(n != 0){
        if(n&1) ans = ans*x % _mod;
        x = x*x % _mod;
        n = n >> 1;
    }
    return ans;
  }
  ll imod(ll n){return _mpow(n , _mod-2);}
 public:
  mint(){ _num = 0;_mod=mod; }
  mint(ll num){ _mod = mod; _num = (num+(1LL<<25)*mod) % mod; }
  mint(ll num,ll M){ _mod=M;_num=(num+(1LL<<25)*mod)%_mod; }
  mint(const mint &cp){_num=cp._num;_mod=cp._mod;}
  mint operator= (const ll x){ return set(x); }
  mint operator+ (const ll x){ return mint(_num + (x % _mod) , _mod); }
  mint operator- (const ll x){ return mint(_num - (x % _mod), _mod); }
  mint operator* (const ll x){ return mint(_num * (x % _mod) , _mod); }
  mint operator/ (ll x){ return mint(_num * imod(x) , _mod);}
  mint operator+=(const ll x){ return set(_num + (x % _mod)); }
  mint operator-=(const ll x){ return set(_num - (x % _mod)); }
  mint operator*=(const ll x){ return set(_num * (x % _mod)); }
  mint operator/=(ll x){ return set(_num * imod(x));}
  mint operator+ (const mint &x){ return mint(_num + x._num , _mod); }
  mint operator- (const mint &x){ return mint(_num - x._num , _mod);}
  mint operator* (const mint &x){ return mint(_num * x._num , _mod); }
  mint operator/ (mint x){ return mint(_num * imod(x._num) , _mod);}
  mint operator+=(const mint &x){ return set(_num + x._num); }
  mint operator-=(const mint &x){ return set(_num - x._num); }
  mint operator*=(const mint &x){ return set(_num * x._num); }
  mint operator/=(mint x){ return set(_num * imod(x._num));}

  bool operator<(const mint &x)const{return _num<x._num;}
  bool operator==(const mint &x)const{return _num==x._num;}
  bool operator>(const mint &x)const{return _num>x._num;}

  friend mint operator+(ll x,const mint &m){return mint(m._num + (x % m._mod) , m._mod);}
  friend mint operator-(ll x,const mint &m){return mint( (x % m._mod) - m._num , m._mod);}
  friend mint operator*(ll x,const mint &m){return mint(m._num * (x % m._mod) , m._mod);}
  friend mint operator/(ll x,mint m){return mint(m.imod(m._num) * x , m._mod);}

  explicit operator ll() { return _num; }
  explicit operator int() { return (int)_num; }

  friend ostream& operator<<(ostream &os, const mint &x){ os << x._num; return os; }
  friend istream& operator>>(istream &is, mint &x){ll val; is>>val; x.set(val); return is;}
};
template<typename T> class MAT{
 private:
    int row,col;
    vector<vector<T>> _A;
    MAT set(vector<vector<T>> A){_A = A ; return *this;}
    
 public:
    MAT(){ }
    MAT(int n,int m){
        if(n<1 || m<0){cout << "err Matrix::Matrix" <<endl;exit(1);}
        row = n;
        col = m?m:n;//m=0のとき単位行列を作る
        REP(i,row){
            vector<T> a(col,0.0);
            _A.push_back(a);
        }
        //  値の初期化
        if(m==0) REP(i,n) _A[i][i]=1.0;
    }
    MAT(const MAT &cp){_A=cp._A;row=cp.row;col=cp.col;}
    T* operator[] (int i){return _A[i].data();}
    MAT operator= (vector<vector<T>> x) {return set(x);}
    MAT operator+ (MAT x) {
        if(row!=x.row || col!=x.col){
            cout << "err Matrix::operator+" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row, col);
        REP(i,row) REP(j,col) r[i][j]=_A[i][j]+x[i][j];
        return r;
    }
    MAT operator- (MAT x) {
        if(row!=x.row || col!=x.col){
            cout << "err Matrix::operator-" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row, col);
        REP(i,row) REP(j,col) r[i][j]=_A[i][j]-x[i][j];
        return r;
    }
    MAT operator* (MAT x) {
        if(col!=x.row){
            cout << "err Matrix::operator*" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row, x.col);
        REP(i,row) REP(j,x.col) REP(k,col) r[i][j]+=_A[i][k]*x[k][j];
        return r;
    }
    MAT operator/ (T a){
        MAT r(row,col);
        REP(i,row) REP(j,col) r[i][j]=_A[i][j]/a;
        return r;
    }
    MAT operator^ (ll n){
        if(row!=col){
            cout << "err Matrix::operator^" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row,0),A=*this;
        while(n){
            if(n&1) r *= A;
            A*=A;
            n>>=1;
        }
        return r;
    }
    MAT operator+= (MAT x) {
        if(row!=x.row || col!=x.col){
            cout << "err Matrix::operator+=" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row, col);
        REP(i,row) REP(j,col) r[i][j]=_A[i][j]+x[i][j];
        return set(r._A);
    }
    MAT operator-= (MAT x) {
        if(row!=x.row || col!=x.col){
            cout << "err Matrix::operator-=" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row, col);
        REP(i,row) REP(j,col) r[i][j]=_A[i][j]-x[i][j];
        return set(r._A);
    }
    MAT operator*= (MAT x) {
        if(col!=x.row){
            cout << "err Matrix::operator*" <<endl;
            cout << "  not equal matrix size" <<endl;
            exit(0);
        }
        MAT r(row, x.col);
        REP(i,row) REP(j,x.col) REP(k,col) r[i][j]+=_A[i][k]*x[k][j];
        return set(r._A);
    }
    MAT operator/=(T a){
        MAT r(row,col);
        REP(i,row) REP(j,col) r[i][j]=_A[i][j]/a;
        return r;
    }

    friend MAT operator* (T n,MAT x){
        MAT r(x.row,x.col);
        REP(i,x.row) REP(j,x.col) r[i][j]=n*x[i][j];
        return r;
    }
    friend MAT operator* (MAT x,T n){
        MAT r(x.row,x.col);
        REP(i,x.row) REP(j,x.col) r[i][j]=n*x[i][j];
        return r;
    }
    explicit operator vector<vector<T>>(){return _A;}
    friend ostream &operator<<(ostream &os,const MAT &x){ REP(i,x.row) REP(j,x.col) os<<x._A[i][j]<<" \n"[j==x.col-1]; return os;}
    friend istream &operator>>(istream &is,MAT &x){REP(i,x.row) REP(j,x.col) is>>x._A[i][j];return is;}
    int size_row(){return row;}
    int size_col(){return col;}
    MAT transpose(){
        MAT r(col,row);
        REP(i,col) REP(j,row) r[i][j]=_A[j][i];
        return r;
    }
    MAT inverse(){
        T buf;
        MAT<T> inv_a(row,0);
        vector<vector<T>> a=_A;
        //掃き出し法
        REP(i,row){
            buf=1/a[i][i];
            REP(j,row){
                a[i][j]*=buf;
                inv_a[i][j]*=buf;
            }
            REP(j,row){
                if(i!=j){
                    buf=a[j][i];
                    REP(k,row){
                        a[j][k]-=a[i][k]*buf;
                       inv_a[j][k]-=inv_a[i][k]*buf;
                    }
                }
            }
        }
        return inv_a;
    }
    // O( n^3 ).
    int rank() {
        vector<vector<T>> A=_A;
        const int n = row, m = col;
        int r = 0;
        for(int i = 0; r < n && i < m; ++i) {
            int pivot = r;
            for(int j = r+1; j < n; ++j) if(fabs(A[j][i]) > fabs(A[pivot][i])) pivot = j;
            swap(A[pivot], A[r]);
            if(fabs(A[r][i]) < eps) continue;
            for (int k = m-1; k >= i; --k) A[r][k] /= A[r][i];
            rep(j,r+1,n) rep(k,i,m) A[j][k] -= A[r][k] * A[j][i];
            ++r;
        }
        return r;
    }
};
class UnionFind{//UnionFind木
 private:
    vector<int> Parent,es;
    vector<ll> diff_weight;
 public:
    UnionFind(int N){
        es.resize(N,0);
        Parent.resize(N,-1);
        diff_weight.resize(N,0LL);
    }

    int root(int A){
        if(Parent[A]<0) return A;
        else{ 
            int r = root(Parent[A]);
            diff_weight[A] += diff_weight[Parent[A]]; // 累積和をとる
            return Parent[A]=r;
        }
    }
    bool issame(int A,int B){
        return root(A)==root(B);
    }
    ll weight(int x) {
        root(x); // 経路圧縮
        return diff_weight[x];
    }
    ll diff(int x, int y) {
        return weight(y) - weight(x);
    }
    int size(int A){
        return -Parent[root(A)];
    }
    int eize(int A){
        return es[root(A)];
    }
    bool connect(int A,int B){
        A=root(A); B=root(B);
        if(A==B) return false;
        if(size(A)<size(B)) swap(A,B);
        Parent[A]+=Parent[B];
        es[A]+=es[B]+1;
        Parent[B]=A;
        return true;
    }
    void unite(int A,int B){
        A=root(A); B=root(B);
        if(A==B){ 
            es[A]++;
            return;
        }
        if(size(A)<size(B)) swap(A,B);
        Parent[A]+=Parent[B];
        es[A]+=es[B]+1;
        Parent[B]=A;
        return;
    }
    bool merge(int A, int B, ll w) {
        // x と y それぞれについて、 root との重み差分を補正
        w += weight(A); w -= weight(B); 
        A=root(A); B=root(B);
        if(A==B) return false;
        if(size(A)<size(B)) swap(A,B),w=-w;
        Parent[A]+=Parent[B];
        Parent[B]=A;
        // x が y の親になるので、x と y の差分を diff_weight[y] に記録
        diff_weight[B] = w; 
        return true;
    }
};
class Factorial{//階乗とその逆元を求めて計算に利用するクラス
 private:
    vector<ll> fac;
    vector<ll> ifac;
 public:
    
    Factorial(ll N){
        fac.push_back(1);
        REP(i,N) fac.push_back(fac[i]*(i+1)%mod);
        ifac.resize(N+1);
        ifac[N]=inv_mod(fac[N]);
        REP(i,N) ifac[N-1-i]=(ifac[N-i]*(N-i))%mod;
    }

    ll fact(ll a){return fac[a];}
    ll ifact(ll a){return ifac[a];}

    ll cmb(ll a,ll b){
        if(a==0&&b==0) return 1;
        if(a<b||a<0||b<0) return 0;
        ll tmp =ifact(a-b)*ifact(b)%mod;
        return tmp*fac[a]%mod;
    }
    ll per(ll a,ll b){
        if(a==0&&b==0) return 1;
        if(a<b||a<0||b<0) return 0;
        return fac[a]*ifac[a-b]%mod;
    }
};
class SOSU{
 private:
    vector<int> Prime_Number;
    vector<bool> isp;
 public:
    SOSU(int N){
        isp.resize(N+1,true);
        isp[0]=isp[1]=false;
        rep(i,2,N+1) if(isp[i]){
            Prime_Number.push_back(i);
            for(int j=2*i;j<=N;j+=i) isp[j]=false;
        }
    }
    int operator[](int i){return Prime_Number[i];}
    int size(){return Prime_Number.size();}
    int back(){return Prime_Number.back();}
    bool isPrime(int q){return isp[q];}
};
class Divisor{//素因数分解をしてから約数列挙、分解結果はＰ(底,指数)でpfacにまとめている
  private:
    vector<ll> F;
    vector<P> pfactorize;
  public:
    Divisor(ll N){
        for(ll i = 1; i * i <= N; i++) {
            if(N % i == 0) {
                F.push_back(i);
                if(i * i != N) F.push_back(N / i);
            }
        }
        sort(begin(F), end(F));
		SOSU p(SQRT(N)+1);
    	REP(i,p.size()){
    		pfactorize.push_back(P(p[i],0));
    		while(N%p[i]==0){
    			N/=p[i];
    			pfactorize.back().second++;
    		}
            if(pfactorize.size()==0) pfactorize.pop_back();
    	}
    	if(N>1) pfactorize.push_back(P(N,1));
    }
    int size(){return F.size();}
    vector<P> pfac(){return pfactorize;}
    ll operator[](int k){return F[k];}
};
struct Solutions{
    ll napsack(int kinds,int MAX_W,const vl &weight,const vl &cost){
        vector<vector<ll>> dp(kinds+1,vector<ll>(MAX_W+1,0));
        REP(i,kinds) REP(j,MAX_W+1){
            if(j<weight[i]) dp[i+1][j]=dp[i][j];
            else dp[i+1][j]=max(dp[i][j],dp[i][j-weight[i]]+cost[i]);
        }
        return dp[kinds][MAX_W];
    }
    ll cost_based_napsack(int kinds,int MAX_W,const vl &weight,const vl &cost){
        int MAX_V=0;
        REP(i,kinds) MAX_V=max(MAX_V,(int)cost[i]);
        vvl dp(kinds+1,vl(kinds*MAX_V+1,1LL<<58));
        dp[0][0] = 0;
        REP(i,kinds) REP(j,kinds*MAX_V+1){
            if(j<cost[i]) dp[i+1][j]=dp[i][j];
            else dp[i+1][j] = min(dp[i][j],dp[i][j-cost[i]]+weight[i]);
        }
        int res=0;
        REP(i,kinds*MAX_V+1) if(dp[kinds][i]<=MAX_W) res=i;
        return res;
    }
    ll unlimited_napsack(int kinds,int MAX_W,const vl &weight,const vl &cost){
        vector<vector<ll>> dp(kinds+1,vector<ll>(MAX_W+1,0));
        REP(i,kinds) REP(j,MAX_W+1){
            if(j<weight[i]) dp[i+1][j]=dp[i][j];
            else dp[i+1][j]=max(dp[i][j],dp[i+1][j-weight[i]]+cost[i]);
        }
        return dp[kinds][MAX_W];
    }
    ll huge_napsack(int kinds,ll MAX_W,const vl &weight,const vl &cost){
        int n2=kinds/2;
        vector<P> ps(1<<(kinds/2));
        REP(i,1<<n2){
            ll sw=0,sv=0;
            REP(j,n2){
                if(i>>j&1){
                    sw += weight[j];
                    sv += cost[j];
                }
            }
            ps[i] = P(sw,sv);
        }
        sort(ps.begin(),ps.begin() + (1<<n2) );
        int m=1;
        rep(i,1,1<<n2){
            if(ps[m-1].second<ps[i].second) ps[m++] = ps[i];
        }

        ll res=0;
        REP(i,1<<(kinds-n2)){
            ll sw=0,sv=0;
            REP(j,kinds-n2){
                if((i>>j)&1){
                    sw += weight[n2+j];
                    sv += cost[n2+j];
                }
            }
            if(sw<=MAX_W){
                ll tv = (lower_bound(ps.begin(),ps.begin()+m,P(MAX_W-sw,LLONG_MAX))-1)->second;
                res = max(res,sv+tv);
            }
        }
        return res;
    }
    ll choose_MonN(int N,int M,const vl &cost){
        vvl dp(N+1,vl(M+1,-1LL<<58));
        REP(i,N+1) dp[i][0]=0;
        REP(i,N) REP(j,M){
            if(j>i) break;
            dp[i+1][j+1]=max(dp[i][j+1],dp[i][j]+cost[i]);
        }
        return dp[N][M];
    }
    ll Partition_Func(int n,int k){
        vector<vector<ll>> dp(k+1,vector<ll> (n+1,0));
        dp[0][0]=1;
        rep(i,1,k+1) REP(j,n+1){
            if(j-i>=0) dp[i][j]=(dp[i][j-i]+dp[i-1][j]);
            else dp[i][j]=dp[i-1][j];
        }
        return dp[k][n];
    }
    int LCS(string s,string t){
        int n=s.length(),m=s.length();
        vector<vector<int>> dp(n+1,vector<int>(m+1));
        REP(i,n) REP(j,m){
            if (s[i] == t[j]) dp[i+1][j+1] = dp[i][j] + 1;
            else dp[i+1][j+1] = max(dp[i][j+1], dp[i+1][j]);
        }
        return dp[n][m];
    }
    int LIS(const vector<ll> a){
        int n=a.size();
        ll INF=1LL<<50;
        vector<ll> dp(n+1,INF);
        REP(i,n) *lower_bound(All(dp),a[i])=a[i];
        int k=lower_bound(All(dp),INF)-dp.begin();
        return k;
    }
    
    int max_flow(int s,int t,const vector<vector<P>> &g){
        struct edge_{int to,cap, rev;};
        vector<bool> used(g.size(),false);
        vector<vector<edge_>> G(g.size());
        REP(i,g.size()) REP(j,g[i].size()){
            int from = i, to = g[i][j].second;
            int cap = g[i][j].first;
            G[from].push_back((edge_){to,cap,(int)G[to].size()});
            G[to].push_back((edge_){from,0,(int)G[from].size()-1});
        }
        auto dfs = [&](auto&& f,int v,int t,int fl)->int{
            if(v==t) return fl;
            used[v] = true;
            REP(i,G[v].size()){
                edge_ &e = G[v][i];
                if(!used[e.to] && e.cap>0){
                    int d = f(f, e.to,t,min(fl,e.cap));
                    if(d>0){
                        e.cap -= d;
                        G[e.to][e.rev].cap += d;
                        return d;
                    }
                }
            }
            return 0;
        };
        int flow=0;
        while(1){
            used.assign(used.size(),false);
            int f = dfs(dfs,s,t,INT_MAX);
            if(f==0) return flow;
            flow += f;
        }
    }
    int bipartite_matching_calculate(const Graph &g){
        int V = g.size();
        vi match(V,-1);
        vector<bool> used(V,false);
        auto dfs = [&](auto&& f,int v)->bool{
            used[v]=true;
            REP(i,g[v].size()){
                int u = g[v][i], w = match[u];
                if(w<0 || !used[w] && f(f,w)){
                    match[v]=u;
                    match[u]=v;
                    return true;
                }
            }
            return false;
        };
        int res=0;
        REP(v,V){
            if(match[v] < 0){
                used.assign(V,false);
                if(dfs(dfs,v)) res++;
            }
        }
        return res;
    }
    int bipartite_matching(int l,int r,function<bool(int,int)> F){
        int V = l+r;
        Graph G(V);
        REP(i,l) REP(j,r) if(F(i,j)){
            G[i].push_back(l+j);
            G[l+j].push_back(i);
        }
        return bipartite_matching_calculate(G);
    }
};
struct compress{
    map<ll,int> zip;
    vector<ll> unzip;
    compress(vector<ll> x){
        unzip=x;
        sort(All(x));
        x.erase(unique(All(x)),x.end());
        REP(i,x.size()){
            zip[x[i]]=i;
        }
    }
    int operator[](int k){return zip[unzip[k]];}
};
struct edge{
    int from;int to;ll cost;
    void push(int a,int b,int c){
        from=a;to=b;cost=c;
    }
    bool operator<(const edge& y) const{
        if(cost!=y.cost) return cost<y.cost;
        else if(to!=y.to) return to<y.to;
        else return from<y.from;}
    bool operator>(const edge& y) const{
        if(cost!=y.cost) return cost>y.cost;
        else if(to!=y.to) return to>y.to;
        else return from>y.from;}
    bool operator==(const edge& y) const{return *this==y;}
};
class lca {
  public:
    const int n = 0;
    const int log2_n = 0;
    std::vector<std::vector<int>> parent;
    std::vector<int> depth;

    lca() {}

    lca(const Graph &g, int root)
        : n(g.size()), log2_n(log2(n) + 1), parent(log2_n, std::vector<int>(n)), depth(n) {
        dfs(g, root, -1, 0);
        for (int k = 0; k + 1 < log2_n; k++) {
            for (int v = 0; v < (int)g.size(); v++) {
                if (parent[k][v] < 0)
                    parent[k + 1][v] = -1;
                else
                    parent[k + 1][v] = parent[k][parent[k][v]];
            }
        }
    }

    void dfs(const Graph &g, int v, int p, int d) {
        parent[0][v] = p;
        depth[v] = d;
        REP(j,g[v].size()) {
            if (g[v][j] != p) dfs(g, g[v][j], v, d + 1);
        }
    }

    int get(int u, int v) {
        if (depth[u] > depth[v]) std::swap(u, v);
        for (int k = 0; k < log2_n; k++) {
            if ((depth[v] - depth[u]) >> k & 1) {
                v = parent[k][v];
            }
        }
        if (u == v) return u;
        for (int k = log2_n - 1; k >= 0; k--) {
            if (parent[k][u] != parent[k][v]) {
                u = parent[k][u];
                v = parent[k][v];
            }
        }
        return parent[0][u];
    }
};
template <typename T,typename E> //SegmentTree((要素数) int n_,(演算) F f,(更新) G g,(初期値) T d1)
struct SegmentTree{
  typedef function<T(T,T)> F;
  typedef function<T(T,E)> G;
  int n;
  F f;
  G g;
  T d1;
  E d0;
  vector<T> dat;
  SegmentTree(){};
  SegmentTree(int n_,F f,G g,T d1,
	      vector<T> v=vector<T>()):
    f(f),g(g),d1(d1){
    init(n_);
    if(n_==(int)v.size()) build(n_,v);
  }
  void init(int n_){
    n=1;
    while(n<n_) n*=2;
    dat.clear();
    dat.resize(2*n-1,d1);
  }
  void build(int n_, vector<T> v){
    for(int i=0;i<n_;i++) dat[i+n-1]=v[i];
    for(int i=n-2;i>=0;i--)
      dat[i]=f(dat[i*2+1],dat[i*2+2]);
  }
  void update(int k,E a){
    k+=n-1;
    dat[k]=g(dat[k],a);
    while(k>0){
      k=(k-1)/2;
      dat[k]=f(dat[k*2+1],dat[k*2+2]);
    }
  }
  inline T query(int a,int b){
    T vl=d1,vr=d1;
    for(int l=a+n,r=b+n;l<r;l>>=1,r>>=1) {
      if(l&1) vl=f(vl,dat[(l++)-1]);
      if(r&1) vr=f(dat[(--r)-1],vr);
    }
    return f(vl,vr);
  }
  
};
struct GP{
    double x,y;
    GP(double X=0,double Y=0):x(X),y(Y){}
    GP(const GP &cp){
        x=cp.x;y=cp.y;
    }
    GP& set(double X,double Y){
        x=X;y=Y;
        return *this;
    }
    GP& operator=(const GP &p){return set(p.x , p.y);}
    GP operator+(const GP &p)const{return GP(x+p.x , y+p.y);}
    GP operator-(const GP &p)const{return GP(x-p.x , y-p.y);}
    GP operator*(double s)const{return GP(x*s,y*s);}
    GP operator/(double s)const{return GP(x/s,y/s);}
    friend GP operator*(double s,const GP &r){return GP(r.x*s,r.y*s);}
    friend GP operator/(double s,const GP &r){return GP(r.x/s,r.y/s);}
    double operator*(GP p)const{return dot(p);}
    double operator^(GP p)const{return cross(p);}
    GP& operator+=(const GP &p){return set(x+p.x , y+p.y);}
    GP& operator-=(const GP &p){return set(x-p.x , y-p.y);}
    GP& operator*=(double s){return set(x*s,y*s);}
    GP& operator/=(double s){return set(x/s,y/s);}
    double dot(const GP &p)const{return x*p.x + y*p.y;} 
    double cross(const GP &p)const{return x*p.y-p.x*y;}
    int ort()const{
        if(fabs(x)<eps && fabs(y)<eps) return 0;
        if(y>0) return (x>0)?1:2;
        else return (x<=0)?3:4;
    }
    bool operator<(const GP &p)const{
        int o=ort(),po=p.ort();
        if(o!=po) return o<po;
        else{
            double cr=cross(p);
            if(cr==0){
                double d=norm(),pd=p.norm();
                return d<pd;
            }
            return cr>0;
        }
    }
    friend istream& operator>>(istream &is, GP &x){double valX,valY; is>>valX>>valY; x.set(valX,valY); return is;}
    friend ostream& operator<<(ostream &os, const GP &v){ os << v.x<<" "<<v.y<<endl; return os; }
    double norm2()const{return x*x+y*y;}
    double norm()const{return sqrt(norm2());}
    double rad()const{
        if(x==0){
            if(y>0) return PI/2.0;
            if(y==0) return -1.0;
            if(y<0) return -PI/2.0;
        }else{
            if(y==0){
                if(x<0) return PI;
                if(x>0) return 0.0;
            }
            else return atan2(y,x);
        }
    }
    void rotate(double theta){
        double X=x,Y=y;
        x=cos(theta)*X-sin(theta)*Y;
        y=sin(theta)*X+cos(theta)*Y;
    }
};
struct Line{
    GP A,B;
    Line(GP p1=GP(0,0),GP p2=GP(1,1)):A(p1),B(p2){}
    double len()const{return (B-A).norm();}
    GP shortest_point(const GP &x)const{
        double a=B.x-A.x,b=B.y-A.y;
        double t=-(a*(A.x-x.x)+b*(A.y-x.y))/(a*a+b*b);
        return GP(a*t+A.x,b*t+A.y);
    }
    double dist(const GP &x)const{
        return (x-shortest_point(x)).norm();
    }
    bool online(const GP &x)const{
        return ((B-A)^(x-A))==0&&(x-A).norm()<=len()&&(x-B).norm()<=len();
    }
    
    friend istream& operator>>(istream &is, Line &x){GP X,Y; is>>X>>Y;x.A=X;x.B=Y;return is;}
    friend ostream& operator<<(ostream &os, const Line &x){ os << x.A << x.B; return os; }
};
struct Triangle{
    GP a,b,c;
    Triangle(GP p1=GP(0,0),GP p2=GP(1,0),GP p3=(0,1)):a(p1),b(p2),c(p3){}

    double A()const{double res=fabs((b-a).rad()-(c-a).rad());if(res>=PI) res-=PI;return res;}
    double B()const{double res=fabs((a-b).rad()-(c-b).rad());if(res>=PI) res-=PI;return res;}
    double C()const{double res=fabs((a-c).rad()-(b-c).rad());if(res>=PI) res-=PI;return res;}
    Line AB()const{return Line(a,b);}
    Line BC()const{return Line(b,c);}
    Line CA()const{return Line(c,a);}

    GP G()const{return (a+b+c)/3;}
    GP O()const{return (sin(2*A())*a+sin(2*B())*b+sin(2*C())*c)/(sin(2*A())+sin(2*B())+sin(2*C()));}
    double R()const{return a.norm()/(2*sin(A()));}
    GP I()const{return (a.norm()*a+b.norm()*b+c.norm()*c)/(a.norm()+b.norm()+c.norm());}
    double r()const{return 2*area()/((a-b).norm()+(b-c).norm()+(c-a).norm());}
    GP H()const{return (tan(A())*a+tan(B())*b+tan(C())*c)/(tan(A())+tan(B())+tan(C()));}
    double AH()const{return 2*R()*cos(A());}
    double BH()const{return 2*R()*cos(B());}
    double CH()const{return 2*R()*cos(C());}
    double area()const{return ((b-a)^(c-a))/2;}

    bool inside(const GP &x)const{
        return ((b-a)^(x-a))*((c-a)^(x-a))<0&&
               ((a-b)^(x-b))*((c-b)^(x-b))<0&&
               ((a-c)^(x-c))*((b-c)^(x-c))<0;
    }
    bool online(const GP &x)const{return AB().online(x)||BC().online(x)||CA().online(x);}
    bool outside(const GP &x)const{return (!inside(x)&&!online(x));}

    friend istream& operator>>(istream &is, Triangle &x){GP X,Y,Z; is>>X>>Y>>Z; x.a=X;x.b=Y;x.c=Z;return is;}
    friend ostream& operator<<(ostream &os, const Triangle &v){ os << v.a << v.b; return os; }
};
struct rational {
  ll p, q;
  void normalize() { // keep q positive
    if (q < 0) p *= -1, q *= -1;
    ll d = gcd(p < 0 ? -p : p, q);
    if (d == 0) p = 0,  q = 1;
    else        p /= d, q /= d;
  }
  rational(ll p, ll q = 1) : p(p), q(q) {
    normalize();
  }
  rational &operator+=(const rational &a){p = a.q * p + a.p * q; q = a.q * q; normalize();return *this;}
  rational &operator-=(const rational &a){p = a.q * p - a.p * q; q = a.q * q; normalize();return *this;}
  rational &operator*=(const rational &a){p *= a.p; q *= a.q;normalize();return *this;}
  rational &operator/=(const rational &a){p *= a.q; q *= a.p; normalize();return *this;}
  rational &operator-(){p *= -1;return *this;}
  friend rational operator+(const rational &a, const rational &b){return rational(a) += b;}
  friend rational operator*(const rational &a, const rational &b){return rational(a) *= b;}
  friend rational operator-(const rational &a, const rational &b){return rational(a)-=b;}
  friend rational operator/(const rational &a, const rational &b){return rational(a) /= b;}
  friend bool operator<(const rational &a, const rational &b){ // avoid overflow 
    return (long double) a.p * b.q < (long double) a.q * b.p;
  }
  friend bool operator<=(const rational &a, const rational &b){return !(b < a);}
  friend bool operator>(const rational &a, const rational &b){return b < a;}
  friend bool operator>=(const rational &a, const rational &b){return !(a < b);}
  friend bool operator==(const rational &a, const rational &b){return !(a < b) && !(b < a);}
  friend bool operator!=(const rational &a, const rational &b){return (a < b) || (b < a);}
  friend ostream& operator<<(ostream &os, const rational &x){ printf("%.16f",(double)x.p/(double)x.q); return os; }
  friend istream& operator>>(istream &is, rational &x){is>>x.p>>x.q; x.normalize(); return is;}
};



int main(){
    cin.tie(nullptr);
    ios::sync_with_stdio(false);
    solve();
    return 0;
}
void solve(){
    int N;cin>>N;
    vector<int> S(1<<N);
    multiset<int> makable;
    REP(i, 1<<N) cin >> S[i],makable.insert(S[i]);
    priority_queue<int> V;
    auto itr=makable.end();itr--;
    V.push(*itr);
    makable.erase(itr);
    bool ok=true;
    REP(i,N){
        priority_queue<int> tmp=V;
        while(!tmp.empty()){
            int slime=tmp.top();tmp.pop();
            auto it=makable.lower_bound(slime);
            if(it==makable.begin()){
                ok=false;break;
            }
            it--;
            V.push(*it);
            makable.erase(it);
        }
        if(!ok) break;
    }
    Yn(ok);
}

Measured memory_bytes: 15220