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 JAVA):

Solution:
import java.io.ByteArrayInputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.InputMismatchException;


public class Main {
	static InputStream is;
	static PrintWriter out;
	static String INPUT = "";
	
	static int Z = 63;
	
	static long[] tob(char[] s)
	{
		long[] ret = new long[Z];
		for(int i = 0;i < s.length;i++){
			ret[i>>>6] |= (long)s[s.length-1-i]-'0'<<i;
		}
		return ret;
	}
	
	static void solve()
	{
		int n = ni();
		long[] X = tob(ns().toCharArray());
		long[][] v = new long[n][];
		for(int i = 0;i < n;i++){
			v[i] = tob(ns().toCharArray());
		}
		
		for(int i = 1;i < n;i++){
			v[0] = gcd(v[0], v[i]);
		}
		
		out.println(count(v[0], X, 998244353));
	}
	
	static long count(long[] b, long[] x, int mod)
	{
		int n = x.length;
		int hb = height(b);
		assert hb > 0;
		long ret = 0;
		long[] o = new long[x.length];
		for(int i = n*64-1;i >= hb-1;i--){
			ret = ret * 2 % mod;
			if(x[i>>>6]<<~i<0){
				ret++;
			}
			if((x[i>>>6]^o[i>>>6])<<~i<0){
				xor(b, 0, hb, o, i-hb+1);
			}
		}
		ret++;
		for(int i = n-1;i >= 0;i--){
			int comp = Long.compareUnsigned(x[i], o[i]);
			if(comp < 0){
				ret--;
				break;
			}
			if(comp > 0){
				break;
			}
		}
		return (int)(ret%mod);
	}
	
	static int height(long[] a)
	{
		for(int i = a.length-1;i >= 0;i--){
			if(a[i] != 0)return i*64+64-Long.numberOfLeadingZeros(a[i]);
		}
		return 0;
	}
	
	static long[] gcd(long[] a, long[] b)
	{
		int ha = height(a);
		int hb = height(b);
		while(hb != 0){
			if(ha >= hb)xor(b, 0, hb, a, ha-hb);
			long[] d = a; a = b; b = d;
			ha = hb; hb = height(b);
		}
		return a;
	}
	
	public static void xor(long[] from, int fl, int fr, long[] to, int tl)
	{
		if(fl >= fr)return;
		if(!(fl >= 0 && fl < from.length<<6))throw new RuntimeException();
		if(!(fr >= 0 && fr <= from.length<<6))throw new RuntimeException();
		if(!(tl >= 0 && tl < to.length<<6))throw new RuntimeException();
		if(!(tl+(fr-fl) >= 0 && tl+(fr-fl) <= to.length<<6))throw new RuntimeException();
		
		int tr = tl+(fr-fl);
		if(tl>>>6 == tr-1>>>6){
			if(fl>>>6 == fr-1>>>6){
				to[tl>>>6] ^= from[fl>>>6]<<-fr>>>fl-fr<<tl;
			}else{
				to[tl>>>6] ^= from[fl>>>6]>>>fl<<tl ^ from[(fl>>>6)+1]<<-fr>>>-fr<<tr-fr;
			}
		}else if((fl&63) < (tl&63)){
			to[tl>>>6] ^= from[fl>>>6]>>>fl<<tl;
			for(int i = (tl>>>6)+1, j = (fl>>>6);i <= (tr-1>>>6)-1;i++,j++){
				to[i] ^= from[j]>>>fl-tl ^ from[j+1]<<-(fl-tl);
			}
			if((fr-1&63) < (tr-1&63)){
				to[tr-1>>>6] ^= from[(fr-1>>>6)-1]>>>fl-tl ^ from[fr-1>>>6]<<-fr>>>-tr;
			}else{
				to[tr-1>>>6] ^= from[fr-1>>>6]<<-fr>>>-tr;
			}
		}else if((fl&63) > (tl&63)){
			to[tl>>>6] ^= from[fl>>>6]>>>fl<<tl ^ from[(fl>>>6)+1]<<-(fl-tl);
			for(int i = (tl>>>6)+1, j = (fl>>>6)+1;i <= (tr-1>>>6)-1;i++,j++){
				to[i] ^= from[j]>>>fl-tl ^ from[j+1]<<-(fl-tl);
			}
			if((fr-1&63) > (tr-1&63)){
				to[tr-1>>>6] ^= from[fr-1>>>6]<<-fr>>>-tr;
			}else{
				to[tr-1>>>6] ^= from[(fr-1>>>6)-1]>>>fl-tl ^ from[fr-1>>>6]<<-fr>>>-tr;
			}
		}else{
			to[tl>>>6] ^= from[fl>>>6]>>>fl<<tl;
			for(int i = (tl>>>6)+1, j = (fl>>>6)+1;i <= (tr-1>>>6)-1;i++,j++){
				to[i] ^= from[j];
			}
			to[tr-1>>>6] ^= from[fr-1>>>6]<<-fr>>>-tr;
		}
	}
	
//		
//		int hba = -1;
//		for(int i = 3999;i >= 0;i--){
//			if(basis[i]){
//				hba = i;
//				break;
//			}
//		}
//		
//		long ans = 0;
//		int mod = 998244353;
//		boolean[] o = new boolean[4000];
//		for(int i = 3999;i >= hba;i--){
//			if(X[i]){
//				ans += pow(2, i-hba, mod);
//			}
//			if(X[i] != o[i]){
//				for(int j = hba;j >= 0;j--){
//					o[i-hba+j] ^= basis[j];
//				}
//			}
//		}
////		tf(X);
////		tf(o);
//		boolean ok = true;
//		for(int i = hba-1;i >= 0;i--){
//			if(!X[i] && o[i]){
//				ok = false;
//				break;
//			}
//			if(X[i] &&!o[i]){
//				ok = true;
//				break;
//			}
//		}
//		if(ok)ans++;
//		out.println(ans%mod);
//	}
	
	public static void tf(boolean... r)
	{
		for(boolean x : r)System.out.print(x?'#':'.');
		System.out.println();
	}
	
	public static void tf(boolean[]... b)
	{
		for(boolean[] r : b) {
			for(boolean x : r)System.out.print(x?'#':'.');
			System.out.println();
		}
		System.out.println();
	}

	
	public static long pow(long a, long n, long mod) {
		//		a %= mod;
		long ret = 1;
		int x = 63 - Long.numberOfLeadingZeros(n);
		for (; x >= 0; x--) {
			ret = ret * ret % mod;
			if (n << 63 - x < 0)
				ret = ret * a % mod;
		}
		return ret;
	}

	
	public static void main(String[] args) throws Exception
	{
		long S = System.currentTimeMillis();
		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
		out = new PrintWriter(System.out);
		
		solve();
		out.flush();
		long G = System.currentTimeMillis();
		tr(G-S+"ms");
	}
	
	private static boolean eof()
	{
		if(lenbuf == -1)return true;
		int lptr = ptrbuf;
		while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
		
		try {
			is.mark(1000);
			while(true){
				int b = is.read();
				if(b == -1){
					is.reset();
					return true;
				}else if(!isSpaceChar(b)){
					is.reset();
					return false;
				}
			}
		} catch (IOException e) {
			return true;
		}
	}
	
	private static byte[] inbuf = new byte[1024];
	static int lenbuf = 0, ptrbuf = 0;
	
	private static int readByte()
	{
		if(lenbuf == -1)throw new InputMismatchException();
		if(ptrbuf >= lenbuf){
			ptrbuf = 0;
			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
			if(lenbuf <= 0)return -1;
		}
		return inbuf[ptrbuf++];
	}
	
	private static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
//	private static boolean isSpaceChar(int c) { return !(c >= 32 && c <= 126); }
	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
	
	private static double nd() { return Double.parseDouble(ns()); }
	private static char nc() { return (char)skip(); }
	
	private static String ns()
	{
		int b = skip();
		StringBuilder sb = new StringBuilder();
		while(!(isSpaceChar(b))){
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	
	private static char[] ns(int n)
	{
		char[] buf = new char[n];
		int b = skip(), p = 0;
		while(p < n && !(isSpaceChar(b))){
			buf[p++] = (char)b;
			b = readByte();
		}
		return n == p ? buf : Arrays.copyOf(buf, p);
	}
	
	private static char[][] nm(int n, int m)
	{
		char[][] map = new char[n][];
		for(int i = 0;i < n;i++)map[i] = ns(m);
		return map;
	}
	
	private static int[] na(int n)
	{
		int[] a = new int[n];
		for(int i = 0;i < n;i++)a[i] = ni();
		return a;
	}
	
	private static int ni()
	{
		int num = 0, b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static long nl()
	{
		long num = 0;
		int b;
		boolean minus = false;
		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
		if(b == '-'){
			minus = true;
			b = readByte();
		}
		
		while(true){
			if(b >= '0' && b <= '9'){
				num = num * 10 + (b - '0');
			}else{
				return minus ? -num : num;
			}
			b = readByte();
		}
	}
	
	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
}