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=list(map(int,input().split()))
if x==1 or x==2*N-1:
    print('No')
else:
    print('Yes')
    a=[]
    if N==2:
        a=[1,2,3]
    elif x==2:
        n=5
        for i in range(2*N-4):
            if i==N-2:
                a.append('\n'.join(map(str,[x+1,x,x-1,x+2])))
            else:
                a.append(n)
                n+=1
    else:
        n=5 if x==3 else 1
        for i in range(2*N-4):
            if i==N-2:
                a.append('\n'.join(map(str,[x-1,x,x+1,x-2])))
            else:
                a.append(n)
                n=x+2 if n==x-3 else n+1
    for i in range(len(a)):
        print(a[i])