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
from statistics import median

sys.setrecursionlimit(10 ** 7)
input = sys.stdin.readline
f_inf = float('inf')
mod = 10 ** 9 + 7
res = 0


def resolve():
    def dfs(S):
        """ デバッグ用 """
        global res
        if len(S) == 1:
            res = S[0]
            return

        tmp = []
        for i in range(1, len(S) - 1):
            t = median([S[i - 1], S[i], S[i + 1]])
            tmp.append(t)
        dfs(tmp)

    n, x = map(int, input().split())
    if x == 1 or x == 2 * n - 1:
        print("No")
        exit()
    n = 2 * n - 1
    ans = [0] * n
    num = x
    for i in range(n // 2, n):
        ans[i] = num
        num += 1
        if num > n:
            num = 1
    for i in range(n // 2):
        ans[i] = num
        num += 1
        if num > n:
            num = 1
    print("Yes")
    print(*ans, sep="\n")
    # dfs(ans)
    # print(res == x)


if __name__ == '__main__':
    resolve()