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 = [ int(v) for v in input().split() ]



if x in [1,2*n-1]:
    print("No")
else:
    print("Yes")
    ans_list = [ None for i in range(2*n-1) ]
    num_list = [ i+1 for i in range(2*n-1) ]

    for i in range(3):
        ans_list[n-2+i] = x-1+i
        num_list[x-2+i] = None
    if x > n:
        ans_list[n+1] = 1
        num_list[0] = None
    elif x < n:
        ans_list[n-3] = 2*n-1
        num_list[2*n-2] = None
    
    num_list = [ i for i in num_list if i != None ]

    t = 0
    for i in ans_list:
        if i == None:
            print(num_list[t])
            t += 1
        else:
            print(i)