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 PYTHON3):

Solution:
from itertools import permutations
n, x = map(int, input().split())
if x == 1 or x == 2 * n - 1:
    print('No')
else:
    print('Yes')
    if n == 2:
        ans = [1, 2, 3]
    else:
        if x == 2:
            ans = [i for i in range(1, x - 1)] \
                  + [i for i in range(x + 3, 2 * n)]
            ans = ans[:n-3] + [x+2, x-1, x, x+1] + ans[n-3:]
        else:
            ans = [i for i in range(1, x - 2)] \
                  + [i for i in range(x + 2, 2 * n)]
            ans = ans[:n-3] + [x-2, x+1, x, x-1] + ans[n-3:]

    for i in range(2 * n - 1):
        print(ans[i])