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.stream.*;
import java.util.*;

public class Main {
    static Scanner scanner = new Scanner(System.in);

    public static void main(String[]$) {
        int n = scanner.nextInt();
        int x = scanner.nextInt();
        if (x == 1 || x == 2 * n - 1) {
            System.out.println("No");
        } else {
            System.out.println("Yes");
            if (n == 2) {
                IntStream.range(1, 2 * n).forEach(System.out::println);
            } else {
                int[] ans = new int[2 * n - 1];
                Set<Integer> use = IntStream.range(1, 2 * n).boxed().collect(Collectors.toSet());
                if (x == 2) {
                    ans[n - 2] = 3;
                    ans[n - 1] = 2;
                    ans[n] = 1;
                    ans[n + 1] = 4;
                    use.removeAll(Arrays.asList(1, 2, 3, 4));
                } else {
                    ans[n - 2] = x - 1;
                    ans[n - 1] = x;
                    ans[n] = x + 1;
                    ans[n + 1] = x - 2;
                    use.removeAll(Arrays.asList(x - 2, x - 1, x, x + 1));
                }
                int i = 0;
                for (int j : use) {
                    if (i >= 2 * n - 1) {
                        break;
                    }
                    ans[i] = j;
                    if (++i == n - 2) {
                        i += 4;
                    }
                }
                Arrays.stream(ans).forEach(System.out::println);
            }
        }
    }
}