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.

<image>

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.



<image>

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[] sa = br.readLine().split(" ");
		int n = Integer.parseInt(sa[0]);
		int x = Integer.parseInt(sa[1]);
		br.close();

		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;
		}

		int[] center = {x - 2, x + 1, x, x - 1};
		int min = x - 2;
		if (x == 2) {
			center[0] = x + 2;
			center[1] = x - 1;
			center[3] = x + 1;
			min = x - 1;
		}
		int num = 0;
		for (int i = 0; i < n - 3; i++) {
			num++;
			if (num == min) {
				num += 4;
			}
			System.out.println(num);
		}
		for (int i = 0; i < 4; i++) {
			System.out.println(center[i]);
		}
		for (int i = 0; i < n - 2; i++) {
			num++;
			if (num == min) {
				num += 4;
			}
			System.out.println(num);
		}
	}
}