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 CPP):

Solution:
#include <stdio.h>

int main(void) {
  int i, n, x, now = 1;
  scanf("%d%d", &n, &x);
  if(x == 1 || x == 2 * n - 1) {
    printf("No");
    return 0;
  }
  int a[2 * n - 1];
  a[n - 1] = x, a[n - 2] = x - 1, a[n] = x + 1;
  for(i = 0; i < n - 2; ++i) {
    if(now == x - 1) now = x + 2;
    a[i] = now++;
  }
  for(i = n + 1; i < 2 * n - 1; ++i) {
    if(now == x - 1) now = x + 2;
    a[i] = now++;
  }
  printf("Yes\n");
  for(i = 0; i < 2 * n - 1; ++i) printf("%d\n", a[i]);
  return 0;
}