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 N == 2:
    if x != N:
        print("No")
        exit(0)
else:
    if abs(N - x) > N - 2:
        print("No")
        exit(0)
print("Yes")

if N == 2:
    for i in range(3):
        print(i+1)
else:
    left = []
    right = []
    if x < 2 * N - 2:
        ans = [x+2, x-1, x, x+1]
        s = [i for i in range(1, 2*N) if i < x - 1 or i > x+2]
        for i in range((2*N - 5) // 2):
            print(s.pop())
        for i in range(4):
            print(ans[i])
        for i in range(((2*N - 5) // 2) + 1):
            print(s.pop())

    else:
        ans = [x-2, x+1, x, x-1]
        s = [i for i in range(1, 2*N) if i < x - 2 or i > x+1]
        for i in range((2*N - 5) // 2):
            print(s.pop())
        for i in range(4):
            print(ans[i])
        for i in range(((2*N - 5) // 2) + 1):
            print(s.pop())