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.

<image>

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.



<image>

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
input = sys.stdin.readline
sys.setrecursionlimit(10 ** 7)

N,X = map(int,input().split())

if X in [1,2*N-1]:
    print('No')
    exit()

if N == 2:
    print('Yes')
    print(*[1,2,3],sep='\n')
    exit()

rest = set(range(1,2*N))
answer = [None] * (2*N-1)
center = N-1
if X <= 2*N-3:
    answer[center-2:center+2] = X+2,X-1,X,X+1
else:
    answer[center-2:center+2] = X-2,X+1,X,X-1
rest -= set(answer[center-2:center+2])
for i in range(2*N-1):
    if answer[i] is None:
        answer[i] = rest.pop()

print('Yes')
print(*answer,sep='\n')