Problem: There are N non-negative integers written on a blackboard. The i-th integer is A_i.

Takahashi can perform the following two kinds of operations any number of times in any order:

* Select one integer written on the board (let this integer be X). Write 2X on the board, without erasing the selected integer.
* Select two integers, possibly the same, written on the board (let these integers be X and Y). Write X XOR Y (XOR stands for bitwise xor) on the blackboard, without erasing the selected integers.



How many different integers not exceeding X can be written on the blackboard? We will also count the integers that are initially written on the board. Since the answer can be extremely large, find the count modulo 998244353.

Constraints

* 1 \leq N \leq 6
* 1 \leq X < 2^{4000}
* 1 \leq A_i < 2^{4000}(1\leq i\leq N)
* All input values are integers.
* X and A_i(1\leq i\leq N) are given in binary notation, with the most significant digit in each of them being 1.

Input

Input is given from Standard Input in the following format:


N X
A_1
:
A_N


Output

Print the number of different integers not exceeding X that can be written on the blackboard.

Examples

Input

3 111
1111
10111
10010


Output

4


Input

4 100100
1011
1110
110101
1010110


Output

37


Input

4 111001100101001
10111110
1001000110
100000101
11110000011


Output

1843


Input

1 111111111111111111111111111111111111111111111111111111111111111
1


Output

466025955

Solution (in CPP):

Solution:
   #include <bits/stdc++.h>

//    #include <boost/multiprecision/cpp_int.hpp>
 #define int long long
 #define inf  1000000007
 #define pa pair<int,int>
 #define ll long long
 #define pal pair<double,double>
 #define ppap pair<pa,int>
  #define PI 3.14159265358979323846
  #define paa pair<int,char>
  #define  mp make_pair
  #define  pb push_back
  #define EPS (1e-10)
                                          
    int dx[8]={0,1,0,-1,1,1,-1,-1};
    int dy[8]={1,0,-1,0,-1,1,1,-1};
                                            using namespace std;
                                   			class pa3{
                                            	public:
                                            	int x;
                                   				int y,z;
                                            	pa3(int x=0,int y=0,int z=0):x(x),y(y),z(z) {}
                                            	bool operator < (const pa3 &p) const{
                                            		if(x!=p.x) return x<p.x;
                                            		if(y!=p.y) return y<p.y;
                                            		 return z<p.z;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                   				bool operator > (const pa3 &p) const{
                                            		if(x!=p.x) return x>p.x;
                                            		if(y!=p.y) return y>p.y;
                                            		 return z>p.z;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator == (const pa3 &p) const{
                                            		return x==p.x && y==p.y && z==p.z;
                                            	}
                                            		bool operator != (const pa3 &p) const{
                                            			return !( x==p.x && y==p.y && z==p.z);
                                            	}
                                            
                                            };
                                            
                                            class pa4{
                                            	public:
                                            	int x;
                                            	int y,z,w;
                                            	pa4(int x=0,int y=0,int z=0,int w=0):x(x),y(y),z(z),w(w) {}
                                            	bool operator < (const pa4 &p) const{
                                            		if(x!=p.x) return x<p.x;
                                            		if(y!=p.y) return y<p.y;
                                            		if(z!=p.z)return z<p.z;
                                            		return w<p.w;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator > (const pa4 &p) const{
                                            		if(x!=p.x) return x>p.x;
                                            		if(y!=p.y) return y>p.y;
                                            		if(z!=p.z)return z>p.z;
                                            		return w>p.w;
                                            		//return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator == (const pa4 &p) const{
                                            		return x==p.x && y==p.y && z==p.z &&w==p.w;
                                            	}
                                            		
                                            
                                            };
                                            class pa2{
                                            	public:
                                            	int x,y;
                                            	pa2(int x=0,int y=0):x(x),y(y) {}
                                            	pa2 operator + (pa2 p) {return pa2(x+p.x,y+p.y);}
                                            	pa2 operator - (pa2 p) {return pa2(x-p.x,y-p.y);}
                                            	bool operator < (const pa2 &p) const{
                                            		return y != p.y ? y<p.y: x<p.x;
                                            	}
                                            	bool operator > (const pa2 &p) const{
                                            		return x != p.x ? x<p.x: y<p.y;
                                            	}
                                            	bool operator == (const pa2 &p) const{
                                            		return abs(x-p.x)==0 && abs(y-p.y)==0;
                                            	}
                                            	bool operator != (const pa2 &p) const{
                                            		return !(abs(x-p.x)==0 && abs(y-p.y)==0);
                                            	}
                                            		
                                            
                                            };
                                            

                      
                                string itos( int i ) {
                                ostringstream s ;
                                s << i ;
                                return s.str() ;
                                }
                                 
                                int gcd(int v,int b){
                                	if(v>b) return gcd(b,v);
                                	if(v==b) return b;
                                	if(b%v==0) return v;
                                	return gcd(v,b%v);
                                }
                 
                            
                                int mod;
int extgcd(int a, int b, int &x, int &y) {
    if (b == 0) {
        x = 1;
        y = 0;
        return a;
    }
    int d = extgcd(b, a%b, y, x);
    y -= a/b * x;
    return d;
}
pa operator+(const pa & l,const pa & r) {   
    return {l.first+r.first,l.second+r.second};                                    
}    
pa operator-(const pa & l,const pa & r) {   
    return {l.first-r.first,l.second-r.second};                                    
}  
                                
                int pr[1200010];
                int inv[1200010];
                
                int beki(int wa,int rr,int warukazu){
                	if(rr==0) return 1%warukazu;
                	if(rr==1) return wa%warukazu;
                	wa%=warukazu;
                	if(rr%2==1) return ((ll)beki(wa,rr-1,warukazu)*(ll)wa)%warukazu;
                	ll zx=beki(wa,rr/2,warukazu);
                	return (zx*zx)%warukazu;
                }

                
    			int comb(int nn,int rr){
    				if(rr<0 || rr>nn || nn<0) return 0;
    				int r=pr[nn]*inv[rr];
    				r%=mod;
    				r*=inv[nn-rr];
    				r%=mod;
    				return r;
    			}
                
                void gya(int ert){
                	pr[0]=1;
                	for(int i=1;i<=ert;i++){
                		pr[i]=(pr[i-1]*i)%mod;
                	}
                		inv[ert]=beki(pr[ert],mod-2,mod);
                	for(int i=ert-1;i>=0;i--){
                		inv[i]=inv[i+1]*(i+1)%mod;
                	}
                }
                
              //   cin.tie(0);
    		//	ios::sync_with_stdio(false);
    			//priority_queue<pa3,vector<pa3>,greater<pa3>> pq;            
                 //sort(ve.begin(),ve.end(),greater<int>());
//    mt19937(clock_per_sec);
  //  mt19937_64 rng(chrono::steady_clock::now().time_since_epoch().count()) ;
      
       
       
                               //----------------kokomade tenpure------------


string gcd(string s,string t){
	while(s.back()=='0')s.pop_back();
	while(t.back()=='0')t.pop_back();
	if(s==t) return s;
	
	if(s.length()<t.length()){
		return gcd(t,s);
	}
	if(s.length()==t.length()){
		int l=t.length();
		for(int i=0;i<l;i++)if(s[i]=='1'){
			if(t[i]=='1')t[i]='0';
			else t[i]='1';
		}
		return gcd(s,t);
	}
	{
		int ls=s.length();
		int lt=t.length();
		
		for(int i=0;i<lt;i++)if(t[i]=='1'){
			if(s[i+ls-lt]=='1')s[i+ls-lt]--;
			else s[i+ls-lt]++;
		}
		
		bool bo=0;
		for(auto c:s)if(c=='1')bo=1;
		if(!bo) return t;
		return gcd(s,t);
	
	}
}

 signed main(){
 	

    	       cin.tie(0);
   			ios::sync_with_stdio(false);
int n;
 	string x;
 	cin>>n>>x;
 	reverse(x.begin(),x.end());
 	string s;
 	cin>>s;
 	reverse(s.begin(),s.end());
 	for(int i=1;i<n;i++){
 		string ss;
 		cin>>ss;
 		
 	reverse(ss.begin(),ss.end());
 		s=gcd(s,ss);
 //		cout<<i<<" "<<s<<endl;
 	}
 //	cout<<s<<endl;
 	mod=998244353;
 	int ans=0;
 	
 	int l=x.length();
 	
 	int r=s.length();
 	
 	if(l<r){
 		cout<<1<<endl;
 		return 0;
 	}
 	

 	
 	string ima="";
 	for(int i=0;i<l-r;i++)ima+="0";
 	ima+=s;
 //	cout<<ans<<endl;
 	for(int i=l-1;i>r-2;i--){
 		if(ima[i]=='0' && x[i]=='0'){
 			goto lll;
 		}
 		else if(ima[i]=='0' &&x[i]=='1'){
 			ans+=beki(2,i-(r-1),mod);
 			ans%=mod;
 		}
 		else if(ima[i]=='1' &&x[i]=='0'){
 			
 		}
 		else{
 			ans+=beki(2,i-(r-1),mod);
 			ans%=mod;
 			goto lll;
 		}
 		
 		for(int j=0;j<r;j++){
 			if(s[j]=='1'){
 				if(ima[i-r+1+j]=='1')ima[i-r+1+j]--;
 				else ima[i-r+1+j]++;
 			}
 		}
 		lll:;
 	//	cout<<ans<<" "<<ima<<" "<<i-r+1<<endl;
 	}
 	
 	bool b=0;
 	for(int i=l-1;i>=0;i--){
 		if(x[i]>ima[i]){
 			b=1;
 			break;
 		}
 		else if(x[i]<ima[i]){
 			goto ll2;
 		}
 	}
 	
 	ans++;
 	ll2:;
 	/*
 	b=0;
 	
 	for(int i=0;i<r;i++){
 		if(s[i]=='1'){
 			if(ima[i]=='0')ima[i]++;
 			else ima[i]--;
 		}
 	}
 	 	for(int i=l-1;i>=0;i--){
 		if(x[i]>ima[i]){
 			b=1;
 			break;
 		}
 		else if(x[i]<ima[i]){
 			break;
 		}
 	}
 	
 	if(b)ans++;
 	*/
 	cout<<ans%mod<<endl;
 	
 	
  	return 0;
  }