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.BufferedReader;
import java.io.InputStreamReader;

public class Main {
    public static void main(String[] args) throws Exception {
        BufferedReader br = new BufferedReader(new InputStreamReader(System.in));
        String line = br.readLine();
        String[] lineValue = line.split(" ");
        int N = Integer.parseInt(lineValue[0]);
        int s = Integer.parseInt(lineValue[1]);
        
        if(s == 2 * N - 1 || s == 1) {
            System.out.println("No");
            return;
        }
        
        System.out.println("Yes");
        
        int outputCount = 1;
        StringBuilder sb = new StringBuilder();
        
        for(int i = 0; i < 2 * N - 1; i++) {
            if(i == N - 2) {
                sb.append(s - 1);
                sb.append("\r\n");
            } else if(i == N - 1) {
                sb.append(s);
                sb.append("\r\n");
            } else if(i == N) {
                sb.append(s + 1);
                sb.append("\r\n");
            } else {
                if(outputCount == s - 1) outputCount += 3;
                sb.append(outputCount++);
                sb.append("\r\n");
            }
        }
        System.out.println(sb.toString());
    }
}