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>
using namespace std;

using ll=long long;
//#define int ll

#define rng(i,a,b) for(int i=int(a);i<int(b);i++)
#define rep(i,b) rng(i,0,b)
#define gnr(i,a,b) for(int i=int(b)-1;i>=int(a);i--)
#define per(i,b) gnr(i,0,b)
#define pb push_back
#define eb emplace_back
#define a first
#define b second
#define bg begin()
#define ed end()
#define all(x) x.bg,x.ed
#ifdef LOCAL
#define dmp(x) cerr<<__LINE__<<" "<<#x<<" "<<x<<endl
#else
#define dmp(x) void(0)
#endif

template<class t,class u> void chmax(t&a,u b){if(a<b)a=b;}
template<class t,class u> void chmin(t&a,u b){if(b<a)a=b;}

template<class t> using vc=vector<t>;
template<class t> using vvc=vc<vc<t>>;

using pi=pair<int,int>;
using vi=vc<int>;

template<class t,class u>
ostream& operator<<(ostream& os,const pair<t,u>& p){
	return os<<"{"<<p.a<<","<<p.b<<"}";
}

template<class t> ostream& operator<<(ostream& os,const vc<t>& v){
	os<<"{";
	for(auto e:v)os<<e<<",";
	return os<<"}";
}

#define mp make_pair
#define mt make_tuple
#define one(x) memset(x,-1,sizeof(x))
#define zero(x) memset(x,0,sizeof(x))
#ifdef LOCAL
void dmpr(ostream&os){os<<endl;}
template<class T,class... Args>
void dmpr(ostream&os,const T&t,const Args&... args){
	os<<t<<" ";
	dmpr(os,args...);
}
#define dmp2(...) dmpr(cerr,"Line:",__LINE__,##__VA_ARGS__)
#else
#define dmp2(...) void(0)
#endif

using uint=unsigned;
using ull=unsigned long long;

template<int i,class T>
void print_tuple(ostream&,const T&){
}

template<int i,class T,class H,class ...Args>
void print_tuple(ostream&os,const T&t){
	if(i)os<<",";
	os<<get<i>(t);
	print_tuple<i+1,T,Args...>(os,t);
}

template<class ...Args>
ostream& operator<<(ostream&os,const tuple<Args...>&t){
	os<<"{";
	print_tuple<0,tuple<Args...>,Args...>(os,t);
	return os<<"}";
}

void print(ll x,int suc=1){
	cout<<x;
	if(suc==1)
		cout<<"\n";
	if(suc==2)
		cout<<" ";
}

ll read(){
	ll i;
	cin>>i;
	return i;
}

vi readvi(int n,int off=0){
	vi v(n);
	rep(i,n)v[i]=read()+off;
	return v;
}

template<class T>
void print(const vector<T>&v,int suc=1){
	rep(i,v.size())
		print(v[i],i==int(v.size())-1?suc:2);
}

string readString(){
	string s;
	cin>>s;
	return s;
}

template<class T>
T sq(const T& t){
	return t*t;
}

//#define CAPITAL
void yes(bool ex=true){
	#ifdef CAPITAL
	cout<<"YES"<<endl;
	#else
	cout<<"Yes"<<endl;
	#endif
	if(ex)exit(0);
}
void no(bool ex=true){
	#ifdef CAPITAL
	cout<<"NO"<<endl;
	#else
	cout<<"No"<<endl;
	#endif
	if(ex)exit(0);
}

constexpr ll ten(int n){
	return n==0?1:ten(n-1)*10;
}

const ll infLL=LLONG_MAX/3;

#ifdef int
const int inf=infLL;
#else
const int inf=INT_MAX/2-100;
#endif

int topbit(signed t){
	return t==0?-1:31-__builtin_clz(t);
}
int topbit(ll t){
	return t==0?-1:63-__builtin_clzll(t);
}
int botbit(signed a){
	return a==0?32:__builtin_ctz(a);
}
int botbit(ll a){
	return a==0?64:__builtin_ctzll(a);
}
int popcount(signed t){
	return __builtin_popcount(t);
}
int popcount(ll t){
	return __builtin_popcountll(t);
}
bool ispow2(int i){
	return i&&(i&-i)==i;
}
int mask(int i){
	return (int(1)<<i)-1;
}

bool inc(int a,int b,int c){
	return a<=b&&b<=c;
}

template<class t> void mkuni(vc<t>&v){
	sort(all(v));
	v.erase(unique(all(v)),v.ed);
}

ll rand_int(ll l, ll r) { //[l, r]
	#ifdef LOCAL
	static mt19937_64 gen;
	#else
    static random_device rd;
    static mt19937_64 gen(rd());
    #endif
    return uniform_int_distribution<ll>(l, r)(gen);
}

template<class t>
int lwb(const vc<t>&v,const t&a){
	return lower_bound(all(v),a)-v.bg;
}

using uint=unsigned;
using ull=unsigned long long;

//initfact();
const uint mod=998244353;
//const uint mod=1000000007;
//uint mod=1;
struct mint{
	uint v;
	mint(ll vv=0){s(vv%mod+mod);}
	mint& s(uint vv){
		v=vv<mod?vv:vv-mod;
		return *this;
	}
	mint operator-()const{return mint()-*this;}
	mint& operator+=(const mint&rhs){return s(v+rhs.v);}
	mint&operator-=(const mint&rhs){return s(v+mod-rhs.v);}
	mint&operator*=(const mint&rhs){
		v=ull(v)*rhs.v%mod;
		return *this;
	}
	mint&operator/=(const mint&rhs){return *this*=rhs.inv();}
	mint operator+(const mint&rhs)const{return mint(*this)+=rhs;}
	mint operator-(const mint&rhs)const{return mint(*this)-=rhs;}
	mint operator*(const mint&rhs)const{return mint(*this)*=rhs;}
	mint operator/(const mint&rhs)const{return mint(*this)/=rhs;}
	mint pow(int n)const{
		mint res(1),x(*this);
		while(n){
			if(n&1)res*=x;
			x*=x;
			n>>=1;
		}
		return res;
	}
	mint inv()const{return pow(mod-2);}
	/*mint inv()const{
		int x,y;
		int g=extgcd(v,mod,x,y);
		assert(g==1);
		if(x<0)x+=mod;
		return mint(x);
	}*/
	friend ostream& operator<<(ostream&os,const mint&m){
		return os<<m.v;
	}
	bool operator<(const mint&r)const{return v<r.v;}
	bool operator==(const mint&r)const{return v==r.v;}
};

const int vmax=(1<<21)+10;
mint fact[vmax],finv[vmax],invs[vmax];
void initfact(){
	fact[0]=1;
	rng(i,1,vmax){
		fact[i]=fact[i-1]*i;
	}
	finv[vmax-1]=fact[vmax-1].inv();
	for(int i=vmax-2;i>=0;i--){
		finv[i]=finv[i+1]*(i+1);
	}
	for(int i=vmax-1;i>=1;i--){
		invs[i]=finv[i]*fact[i-1];
	}
}
mint choose(int n,int k){
	return fact[n]*finv[n-k]*finv[k];
}
mint binom(int a,int b){
	return fact[a+b]*finv[a]*finv[b];
}
mint catalan(int n){
	return binom(n,n)-(n-1>=0?binom(n-1,n+1):0);
}

const int L=4000;
//const int L=5;
//const int L=10;
using B=bitset<L>;

B basis[L];
bool has[L];
int rem=L;

vc<B> qs;

void ad(B b){
	per(i,L){
		if(b[i]){
			if(has[i]){
				b^=basis[i];
			}else{
				rem--;
				has[i]=1;
				basis[i]=b;
				if(i<L-1)
					qs.pb(b<<1);
				break;
			}
		}
	}
}

signed main(){
	cin.tie(0);
	ios::sync_with_stdio(0);
	cout<<fixed<<setprecision(20);
	
	int n;cin>>n;
	B x;cin>>x;
	
	rep(_,n){
		B b;cin>>b;
		qs.pb(b);
	}
	while(rem&&qs.size()){
		B b=qs.back();qs.pop_back();
		ad(b);
	}
	
	per(i,L)if(has[i]){
		dmp(basis[i]);
	}
	
	mint p=1,q;
	B cur;
	per(i,L){
		if(has[i]){
			q*=2;
			if(x[i]){
				if(cur[i]){
					q+=p;
				}else{
					q+=p;
					cur^=basis[i];
				}
			}else{
				if(cur[i]){
					cur^=basis[i];
				}else{
				}
			}
		}else{
			if(x[i]){
				if(cur[i]){
				}else{
					q+=p;
					p=0;
				}
			}else{
				if(cur[i]){
					p=0;
				}else{
				}
			}
		}
	}
	
	cout<<p+q<<endl;
}