Problem: We have a pyramid with N steps, built with blocks. The steps are numbered 1 through N from top to bottom. For each 1≤i≤N, step i consists of 2i-1 blocks aligned horizontally. The pyramid is built so that the blocks at the centers of the steps are aligned vertically.

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A pyramid with N=4 steps

Snuke wrote a permutation of (1, 2, ..., 2N-1) into the blocks of step N. Then, he wrote integers into all remaining blocks, under the following rule:

* The integer written into a block b must be equal to the median of the three integers written into the three blocks directly under b, or to the lower left or lower right of b.



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Writing integers into the blocks

Afterwards, he erased all integers written into the blocks. Now, he only remembers that the integer written into the block of step 1 was x.

Construct a permutation of (1, 2, ..., 2N-1) that could have been written into the blocks of step N, or declare that Snuke's memory is incorrect and such a permutation does not exist.

Constraints

* 2≤N≤10^5
* 1≤x≤2N-1

Input

The input is given from Standard Input in the following format:


N x


Output

If no permutation of (1, 2, ..., 2N-1) could have been written into the blocks of step N, print `No`.

Otherwise, print `Yes` in the first line, then print 2N-1 lines in addition.

The i-th of these 2N-1 lines should contain the i-th element of a possible permutation.

Examples

Input

4 4


Output

Yes
1
6
3
7
4
5
2


Input

2 1


Output

No

Solution (in JAVA):

Solution:
import java.io.*;

public class Main {
    
    public static void main(String[] args) throws IOException {

        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        String[] in1 = br.readLine().split(" ");
        int N = Integer.parseInt(in1[0]);
        int x = Integer.parseInt(in1[1]);
        br.close();

        int[] ans = new int[2*N - 1];
        
        if(x == 1 || x == 2 * N - 1){
            System.out.println("No");
            return;
        }

        System.out.println("Yes");
        
        if(N == 2){
            System.out.println(1);
            System.out.println(2);
            System.out.println(3);
            return;
        }

        boolean[] used = new boolean[2*N];

        ans[N-1] = x;
        used[x] = true;
        if(x == 2){
            ans[N] = 1; used[1] = true;
            ans[N-2] = 3; used[3] = true;
            ans[N+1] = 4; used[4] = true;
        }else{
            ans[N] = x+1; used[x+1] = true;
            ans[N-2] = x-1; used[x-1] = true;
            ans[N+1] = x-2; used[x-2] = true;
        }

        int cnt = 0;
        for(int i = 1; i <= 2 * N - 1; i++){
            if(used[i]) continue;
            while(ans[cnt] != 0) cnt++;
            ans[cnt] = i;
            cnt++;
            used[i] = true;
        }
        for(int i = 0; i < 2 * N - 1; i++){
            System.out.println(ans[i]);
        }
    }
}