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
#import numpy as np
stdin = sys.stdin

ri = lambda: int(rs())
rl = lambda: list(map(int, stdin.readline().split())) # applies to numbers only
rs = lambda: stdin.readline().rstrip()  # ignore trailing spaces

N, x = rl()
if x == 1 or x == 2 * N - 1:
    print('No')
    exit()
answer = [0] * (2*N-1)
mid = N-1
if x <= N:
    answer[mid] = x
    answer[mid-1] = x - 1
    cur = 0
    for y in range(2*N-1, 0, -1):
        while cur == mid or cur == mid-1:
            cur += 1
        if y == x or y == x-1:
            continue
        answer[cur] = y
        cur += 1
else:
    answer[mid] = x
    answer[mid-1] = x + 1
    cur = 0
    for y in range(1, 2*N):
        while cur == mid or cur == mid-1:
            cur += 1
        if y == x or y == x+1:
            continue
        answer[cur] = y
        cur += 1

print('Yes')
print(*answer)
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