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.*;
import java.math.BigInteger;
import java.util.*;

public class Main {
    private FastScanner in;
    private PrintWriter out;

    final int HASH_MOD = (int) 1e9 + 7;

    class MyNumber {
        BitSet value;
        int[] hash;
        int[] firstOne;
        int len;

        MyNumber(BitSet s) {
            value = s;
            int slen = s.length();
            hash = new int[slen + 1];
            for (int i = 0; i < slen; i++) {
                hash[i + 1] = (int) ((hash[i] * 29L + (s.get(s.length() - 1 - i) ? 1 : 0)) % HASH_MOD);
            }
            len = slen;
            firstOne = new int[slen + 1];
            firstOne[slen] = slen;
            for (int i = slen - 1; i >= 0; i--) {
                if (s.get(slen - 1 - i)) {
                    firstOne[i] = i;
                } else {
                    firstOne[i] = firstOne[i + 1];
                }
            }
        }

        MyNumber(String s) {
            this(read(s));
        }

        @Override
        public String toString() {
            return value.toString();
        }
    }

    int MAX;

    MyNumber[] numbers;

    int getSameLen(MyNumber a, MyNumber b) {
        int r = Math.min(a.len, b.len) + 1;
        int l = 0;
        while (r - l > 1) {
            int m = (l + r) >> 1;
            if (a.hash[m] == b.hash[m]) {
                l = m;
            } else {
                r = m;
            }
        }
        return l;
    }

    int calcLenOfDiff(MyNumber a, MyNumber b) {
        if (a.len < b.len) {
            return calcLenOfDiff(b, a);
        }
        int sameLen = getSameLen(a, b);
        if (sameLen == b.len) {
            int firstOne = a.firstOne[sameLen];
            return a.len - firstOne;
        } else {
            return a.len - sameLen;
        }
    }

    List<MyNumber> list = new ArrayList<>();

    BitSet shiftLeft(BitSet x, int shift) {
        if (x.cardinality() == 0) {
            throw new AssertionError();
        }
        int len = x.length();
        BitSet newOne = new BitSet(len + shift);
        for (int i = 0; i < len; i++) {
            if (x.get(i)) {
                newOne.set(i + shift);
            }
        }
        return newOne;
    }

    BitSet xor(BitSet one, BitSet two, int len) {
        BitSet res = new BitSet(len);
        for (int i = 0; i < len; i++) {
            if (one.get(i) != two.get(i)) {
                res.set(i);
            }
        }
        return res;
    }

    void add(MyNumber number) {
        if (numbers[number.len] == null) {
            numbers[number.len] = number;
        }
        for (int i = 0; i < MAX; i++) {
            if (numbers[i] != null) {

                int diffLen = calcLenOfDiff(number, numbers[i]);
                if (diffLen != 0 && numbers[diffLen] == null) {

                    BitSet first = number.value;
                    BitSet second = numbers[i].value;
                    if (number.len > numbers[i].len) {
                        second = shiftLeft(second, number.len - numbers[i].len);
                    } else {
                        first = shiftLeft(first, numbers[i].len - number.len);
                    }
                    MyNumber next = new MyNumber(xor(first, second, diffLen));
                    numbers[diffLen] = next;
                    if (next.len != diffLen) {
                        throw new AssertionError();
                    }
                    list.add(next);
                }

            } else {

                if (i > number.len) {
                    MyNumber next = new MyNumber(shiftLeft(number.value, i - number.len));
                    numbers[i] = next;
                    if (next.len != i) {
                        System.err.println(number + " " + i + " " + next + " " + (i - number.len));
                        throw new AssertionError();
                    }
                    list.add(next);
                }

            }
        }
    }

    MyNumber gcd(MyNumber a, MyNumber b) {
        if (a == null) {
            return b;
        }
        if (a.len < b.len) {
            return gcd(b, a);
        }
        int shift = a.len - b.len;
        BitSet next = new BitSet(a.len);
        for (int i = 0; i < a.len; i++) {
            boolean xx = a.value.get(i);
            if (i >= shift) {
                xx ^= b.value.get(i - shift);
            }
            if (xx) {
                next.set(i);
            }
        }
        if (next.cardinality() == 0) {
            return b;
        }
        return gcd(b, new MyNumber(next));
    }

    BitSet read(String s) {
        BitSet res = new BitSet(s.length());
        for (int i = 0; i < s.length(); i++) {
            if (s.charAt(i) == '1') {
                res.set(s.length() - 1 - i);
            }
        }
        return res;
    }

    private void solve() {
        int n = in.nextInt();
        long START = System.currentTimeMillis();
        BitSet X = read(in.next());
        MAX = X.length() + 5;
        MyNumber GCD = null;
        numbers = new MyNumber[MAX];
        for (int i = 0; i < n; i++) {
            MyNumber number = new MyNumber(in.next());
            GCD = gcd(GCD, number);
//            add(number);
        }
        add(GCD);
        final int mod = 998244353;
        BitSet current = new BitSet();
        int[] pow2 = new int[MAX];
        pow2[0] = 1;
        for (int i = 1; i < MAX; i++) {
            pow2[i] = pow2[i - 1];
            if (numbers[i] != null) {
                pow2[i] *= 2;
                pow2[i] %= mod;
            }
        }
        long result = 0;
        boolean found = true;
        for (int i = MAX - 2; i >= 0; i--) {
            boolean xBit = X.get(i);
            boolean myBit = current.get(i);
            if (numbers[i + 1] != null) {
                if (xBit) {
                    result += pow2[i];
                    if (!myBit) {
                        current.xor(numbers[i + 1].value);
                    }
                } else {
                    if (myBit) {
                        current.xor(numbers[i + 1].value);
                    }
                }
            } else {
                if (xBit != myBit) {
                    if (myBit) {
                    } else {
                        result += pow2[i + 1];
                    }
                    found = false;
                    break;
                }
            }
        }
        if (found) {
            result++;
        }
        out.println(result % mod);
        System.err.println(System.currentTimeMillis() - START);
    }

    private void run() {
        try {
            in = new FastScanner(new File("ARC_084_D.in"));
            out = new PrintWriter(new File("ARC_084_D.out"));

            solve();

            out.close();
        } catch (FileNotFoundException e) {
            e.printStackTrace();
        }
    }

    private void runIO() {
        in = new FastScanner(System.in);
        out = new PrintWriter(System.out);

        solve();

        out.close();
    }

    private class FastScanner {
        BufferedReader br;
        StringTokenizer st;

        FastScanner(File f) {
            try {
                br = new BufferedReader(new FileReader(f));
            } catch (FileNotFoundException e) {
                e.printStackTrace();
            }
        }

        FastScanner(InputStream f) {
            br = new BufferedReader(new InputStreamReader(f));
        }

        String next() {
            while (st == null || !st.hasMoreTokens()) {
                String s = null;
                try {
                    s = br.readLine();
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (s == null)
                    return null;
                st = new StringTokenizer(s);
            }
            return st.nextToken();
        }

        boolean hasMoreTokens() {
            while (st == null || !st.hasMoreTokens()) {
                String s = null;
                try {
                    s = br.readLine();
                } catch (IOException e) {
                    e.printStackTrace();
                }
                if (s == null)
                    return false;
                st = new StringTokenizer(s);
            }
            return true;
        }

        int nextInt() {
            return Integer.parseInt(next());
        }

        long nextLong() {
            return Long.parseLong(next());
        }

        double nextDouble() {
            return Double.parseDouble(next());
        }
    }

    public static void main(String[] args) {
        new Main().runIO();
    }
}