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.util.ArrayDeque;
import java.util.Scanner;

public class Main{
	public static void main(String[] args){
		Scanner sc=new Scanner(System.in);
		while(sc.hasNext()){
			int N = sc.nextInt(), x = sc.nextInt();
			if(x == 1 || x == 2 * N - 1) {
				System.out.println("No");
				break;
			}
			System.out.println("Yes");
			int low = x - 1, high = x + 1;
			ArrayDeque<Integer> deq = new ArrayDeque<Integer>();
			deq.add(x);
			while(low != 0 || high != 2*N) {
				if(low == 0) {
					deq.push(high);
					deq.add(high+1); high += 2;
				}else if(high == 2*N) {
					deq.add(low);
					deq.push(low - 1); low -= 2;
				}else {
					deq.push(low); low--;
					deq.add(high); high++;
				}
			}
			for(int s : deq) System.out.println(s);
		}
	}
}