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:
def main():

    def calc(lst):
        now = lst
        while len(now) > 1:
            tmp = []
            for i in range(len(now) - 2):
                ABC = [now[i], now[i + 1], now[i + 2]]
                ABC.sort()
                tmp.append(ABC[1])
            now = tmp
        return now[0]

    N, x = map(int, input().split())
    if x == 1 or x == 2 * N - 1:
        print ('No')
        return 0
    print ('Yes')
    ans = [(1 + i) for i in range(2 * N - 1)]
    if x < N:
        tmp = N - x + 1
        for i in range((tmp + 1) // 2):
            ans[N - i - 1], ans[N - tmp + i - 1] = ans[N - tmp + i - 1], ans[N - i -1]
    else:
        tmp = x - N + 1
        for i in range((tmp + 1) // 2):
            ans[N + i - 1], ans[N + tmp - i - 1] = ans[N + tmp - i - 1], ans[N + i - 1]
    print (*ans, sep = '\n')
    # print (calc(ans))

if __name__ == '__main__':
    main()