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:
import sys

readline = sys.stdin.readline
MOD = 10 ** 9 + 7
INF = float('INF')
sys.setrecursionlimit(10 ** 5)


def main():
    n, x = map(int, readline().split())
    ans = [0] * (2 * n - 1)

    if x == 1 or x == (2 * n - 1):
        print("No")
        return

    for i in range(1, n + 1):
        if i % 2 == 1:
            ans[i - 1] = (x - 1 + abs(n - i)) % (2 * n - 1) + 1
        else:
            ans[i - 1] = (x - 1 - abs(n - i)) % (2 * n - 1) + 1
    ans[n] = x

    for i in range(n + 1, 2 * n):
        if i % 2 == 0:
            ans[i - 1] = (x - 1 + abs(n - i)) % (2 * n - 1) + 1
        else:
            ans[i - 1] = (x - 1 - abs(n - i)) % (2 * n - 1) + 1
    
    print("Yes")
    print(*ans, sep="\n")


if __name__ == '__main__':
    main()