Problem: You are given a binary string s (each character of this string is either 0 or 1).

Let's denote the cost of string t as the number of occurences of s in t. For example, if s is 11 and t is 111011, then the cost of t is 3.

Let's also denote the Fibonacci strings sequence as follows:

  * F(0) is 0;
  * F(1) is 1;
  * F(i) = F(i - 1) + F(i - 2) if i > 1, where  +  means the concatenation of two strings.



Your task is to calculate the sum of costs of all subsequences of the string F(x). Since answer may be large, calculate it modulo 109 + 7.

Input

The first line contains two integers n and x (1 ≤ n ≤ 100, 0 ≤ x ≤ 100) — the length of s and the index of a Fibonacci string you are interested in, respectively.

The second line contains s — a string consisting of n characters. Each of these characters is either 0 or 1.

Output

Print the only integer — the sum of costs of all subsequences of the string F(x), taken modulo 109 + 7. 

Examples

Input

2 4
11


Output

14


Input

10 100
1010101010


Output

553403224

Solution (in JAVA):

Solution:
import java.util.*;
import java.io.*;
import java.lang.*;
import java.math.*;
public class F {
    public static void main(String[] args) throws Exception {
        BufferedReader bf = new BufferedReader(new InputStreamReader(System.in));
        // Scanner scan = new Scanner(System.in);
        // PrintWriter out = new PrintWriter(new OutputStreamWriter(System.out));
        // int n = Integer.parseInt(bf.readLine());
        StringTokenizer st = new StringTokenizer(bf.readLine());
        // int[] a = new int[n];
        // for(int i=0; i<n; i++) a[i] =  Integer.parseInt(st.nextToken());
        int n = Integer.parseInt(st.nextToken());
        int x = Integer.parseInt(st.nextToken());

        String s = bf.readLine();
        if(x==0) {
          if(s.equals("0"))
            System.out.println("1");
          else System.out.println("0");
          System.exit(0);
        }
        int mod = 1000000007;
        int[][][] dp = new int[n][n][x+1];
        int[][][] dp_begin = new int[n][n][x+1];
        int[][][] dp_end = new int[n][n][x+1];
        int[][][] dp_final = new int[n][n][x+1];
        int[] fibonacci = new int[x+1];
        fibonacci[0] = 1;
        if(x >= 1) fibonacci[1] = 1;
        for(int i=2; i<=x; i++)
          fibonacci[i] = (int)((fibonacci[i-1] + fibonacci[i-2]) % (mod - 1));
        for(int i=0; i<n; i++) {

              if(s.charAt(i) == '1') dp[i][i][1] = 1;
              if(s.charAt(i) == '0') dp[i][i][0] = 1;
              if(s.charAt(i) == '1') dp_begin[i][i][1] = 1;
              if(s.charAt(i) == '0') dp_begin[i][i][0] = 1;
              if(s.charAt(i) == '1') dp_end[i][i][1] = 1;
              if(s.charAt(i) == '0') dp_end[i][i][0] = 1;
              if(s.charAt(i) == '1') dp_final[i][i][1] = 1;
              if(s.charAt(i) == '0') dp_final[i][i][0] = 1;

        }
        int[] exponential = new int[x+1];
        for(int i=0; i<=x; i++) exponential[i] = exp(2, fibonacci[i], mod);
        for(int k=2; k<=x; k++) {
          for(int i=0; i<n; i++) {
            for(int j=i; j<n; j++) {

              int sum = 0;
              long toAdd = 1;
              for(int c=0; c<=j-i+1; c++) {
                toAdd = 1;
                if(c != 0) toAdd *= dp[i][i+c-1][k-1];
                if(c != j-i+1) toAdd *= dp[i+c][j][k-2];
                toAdd %= mod;

                sum = (int)((0L + sum + toAdd) % mod);
              }
              dp[i][j][k] = sum;
            }
          }
        }
        for(int k=2; k<=x; k++) {
          for(int i=0; i<n; i++) {
            for(int j=i; j<n; j++) {

              int sum = 0;
              long toAdd = 1;
              for(int c=0; c<=j-i+1; c++) {
                toAdd = 1;
                if((c != 0) && (c != j-i+1)) {
                  toAdd *= dp_begin[i][i+c-1][k-1];
                  toAdd *= dp[i+c][j][k-2];
                }
                else if(c == 0) {
                  toAdd *= exponential[k-1];
                  toAdd *= dp_begin[i+c][j][k-2];
                }
                else if(c == j-i+1)  {
                  toAdd *= dp_begin[i][i+c-1][k-1];
                }
                toAdd %= mod;
                sum = (int)((0L + sum + toAdd) % mod);
              }
              dp_begin[i][j][k] = sum;
            }
          }
        }
        for(int k=2; k<=x; k++) {
          for(int i=0; i<n; i++) {
            for(int j=i; j<n; j++) {

              int sum = 0;
              long toAdd = 1;
              for(int c=0; c<=j-i+1; c++) {
                toAdd = 1;
                if((c != 0) && (c != j-i+1)) {
                  toAdd *= dp[i][i+c-1][k-1];
                  toAdd *= dp_end[i+c][j][k-2];
                }
                else if(c == 0) {

                  toAdd *= dp_end[i+c][j][k-2];
                }
                else if(c == j-i+1)  {
                  toAdd *= exponential[k-2];
                  toAdd *= dp_end[i][i+c-1][k-1];
                }
                toAdd %= mod;
                sum = (int)((0L + sum + toAdd) % mod);
              }
              dp_end[i][j][k] = sum;
            }
          }
        }
        for(int k=2; k<=x; k++) {
          for(int i=0; i<n; i++) {
            for(int j=i; j<n; j++) {

              int sum = 0;
              long toAdd = 1;
              for(int c=0; c<=j-i+1; c++) {
                toAdd = 1;
                if((c != 0) && (c != j-i+1)) {
                  toAdd *= dp_begin[i][i+c-1][k-1];
                  toAdd *= dp_end[i+c][j][k-2];
                }
                else if(c == 0) {
                  toAdd *= 1L*exponential[k-1]*dp_final[i+c][j][k-2];

                }
                else if(c == j-i+1) {
                  toAdd *= 1L*dp_final[i][i+c-1][k-1]*exponential[k-2];
                }

                toAdd %= mod;
                sum = (int)((0L + sum + toAdd) % mod);
              }
              dp_final[i][j][k] = sum;
            }
          }
        }
        //System.out.println(dp[0][0][3]);
      //  System.out.println("Occurences: " + dp[0][n-1][x]);
        int ans = dp_final[0][n-1][x];

        System.out.println(ans);
        // out.close(); System.exit(0);
    }

    public static int exp(int base, int e, int mod) {
      if(e == 0) return 1;
      if(e == 1) return base;
      int val = exp(base, e/2, mod);
      int ans = (int)(1L*val*val % mod);
      if(e % 2 == 1)
        ans = (int)(1L*ans*base % mod);
      return ans;
    }
}