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 PYTHON3):

Solution:
# 解説見た

def solve(n, x):
    if x in {1, n * 2 - 1}:
        return ['No']

    if n == 2:
        a = ['Yes'] + [1, 2, 3]
        return a

    a = [-1] * (n * 2 - 1)
    a[n - 1] = x

    if x == 2:
        a[n - 2] = n * 2 - 2
        a[n] = 1
        a[n + 1] = n * 2 - 1
    else:
        a[n - 2] = 1
        a[n] = n * 2 - 1
        a[n + 1] = 2

    remains = set(range(1, n * 2)) - set(a)
    for ind, a_ in enumerate(a):
        if a_ == -1:
            a[ind] = remains.pop()

    a = ['Yes'] + a
    return a


if __name__ == '__main__':
    n, x = map(int, input().split())
    print(*solve(n, x), sep='\n')