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;
typedef long long ll;
// typedef __int128 ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
#define FR first
#define SE second
#define MP make_pair
#define PB push_back
#define vc vector
#define db double
#define all(x) (x).begin(),(x).end()
#define sz(x) ((int)(x).size())
#define bin(x) (1ll<<(x))
#define fo(i,l,r) for(int i=(l),I=(r);i<=I;i++)
#define fd(i,r,l) for(int i=(r),I=(l);i>=I;i--)
#define mem(x,val) memset(x,val,sizeof x)
#define Swap(a,b,n) for(int I=0;I<=n;I++) swap(a[I],b[I])
#define PC __builtin_popcountll
#ifdef DEBUG
	#define debug(A,args...) fprintf(stderr,A,##args)
#else
	#define debug(A,args...) printf("")
#endif
#define deb debug("line %d\n",__LINE__)
namespace mine
{
	ll qread()
	{
		ll ans=0,f=1;char c=getchar();
		while(c<'0' or c>'9') {if(c=='-')f=-1;c=getchar();}
		while('0'<=c and c<='9') ans=ans*10+c-'0',c=getchar();
		return ans*f;
	}
	void write(ll num){if(num<0) putchar('-'),num=-num;if(num>=10) write(num/10);putchar('0'+num%10);}
	void write1(ll num){write(num);putchar(' ');}
	void write2(ll num){write(num);putchar('\n');}
	template<typename T>inline bool chmax(T&a,const T&b){return a<b?a=b,1:0;}
	template<typename T>inline bool chmin(T&a,const T&b){return a>b?a=b,1:0;}
	ll gcd(ll x,ll y){return y?gcd(y,x%y):x;}
	bool IN(ll x,ll l,ll r){return l<=x and x<=r;}
	void GG(){puts("-1");exit(0);}

	const db eps=1e-8;
	const int INF=0x3f3f3f3f;
	const int MOD=998244353;
	int mm(const int x){return x>=MOD?x-MOD:x;}
	template<typename T> void add(T &x,const int &y){x=(x+y>=MOD?x+y-MOD:x+y);}
	ll qpower(ll x,ll e,int mod=MOD){ll ans=1;while(e){if(e&1)ans=ans*x%mod;x=x*x%mod;e>>=1;}return ans;}
	ll invm(ll x){return qpower(x,MOD-2);}
	const int MM=1e6+10;
	ll fac[MM],facinv[MM],Inv[MM];ll Comb(int n,int m){return n<0 or n<m?0:fac[n]*facinv[m]%MOD*facinv[n-m]%MOD;}
	void PRE()
	{
		fac[0]=1;fo(i,1,MM-1) fac[i]=fac[i-1]*i%MOD;
		facinv[MM-1]=invm(fac[MM-1]);fd(i,MM-1,1) facinv[i-1]=facinv[i]*i%MOD;
		Inv[1]=1;fo(i,2,MM-1) Inv[i]=(MOD-MOD/i)*Inv[MOD%i]%MOD;
	}
	const int N=4e3+10;
	//------------------FIXED------------------
	typedef bitset<N> bs;
	bool operator <= (bs a,bs b)
	{
		fd(i,N-1,0) if(a[i]!=b[i]) return a[i]<b[i];
		return 1;
	}
	char str[N];int ln;
	void read(bs &ret){ scanf("%s",str+1);ln=strlen(str+1);fo(i,1,ln) ret[ln-i]=(str[i]-'0'); }
	void main()
	{
		int n=qread();bs MX;read(MX);int L=ln;
		bs gcd;read(gcd);int ln1=ln;
		fo(i,2,n)
		{
			bs now;read(now);int ln2=ln;
			while(1)
			{
				if(ln1>ln2) swap(ln1,ln2),swap(gcd,now);
				now^=gcd<<(ln2-ln1);while(ln2 and now[ln2-1]==0) ln2--;if(ln2==0) break;
			}
		}
		if(ln1>L){write(1);return;}//0
		ll ans=0;int z=L-ln1+1;
		bs now;fd(i,L-1,L-z) {ans=(ans*2+MX[i])%MOD;if(MX[i]^now[i]) now^=gcd<<(i-ln1+1);}
		if(now<=MX) add(ans,1);write(ans);
	}
};//变量重名！
signed main()
{
	#ifdef DEBUG
	//freopen("a.in","r",stdin);
	freopen("z.txt","r",stdin);
	//freopen("a.out","w",stdout);
	#endif
	srand(time(0));
	mine::PRE();
	mine::main();
	debug("\n------------------------------------------\nTime: %.2lf s\n",1.0*clock()/CLOCKS_PER_SEC);
}