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:
n,x = map(int, input().split())
if 1 < x < (2*n - 1):
    print("Yes")
    if n == 2:
        for i in range(3):
            print(i+1)
        exit()
    if x != 2:
        ans = [-1 for i in range(2*n-1)]
        k = (n*2-1) //2
        ans[k-1] = x-1
        ans[k] = x
        ans[k+1] = x+1
        ans[k+2] = x-2
        used = set([x-1,x,x+1,x-2])
        insert = iter([i+1 for i in range(n*2-1) if (i+1) not in used])
        for i in range(2*n-1):
            if ans[i] == -1:
                ans[i] = insert.__next__()
        for i in range(n*2-1):
            print(ans[i])
    else:
        ans = [-1 for i in range(2*n-1)]
        k = (n*2-1) // 2
        ans[k-1] = x+2
        ans[k] = x
        ans[k+1] = x-1
        ans[k+2] = x+1
        used = set([x+1,x,x+2,x-1])
        insert = iter([i+1 for i in range(n*2-1) if (i+1) not in used])
        for i in range(2*n-1):
            if ans[i] == -1:
                ans[i] = insert.__next__()
        for i in range(n*2-1):
            print(ans[i])
else:
    print("No")