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 heapq
from operator import itemgetter
from collections import deque, defaultdict
from bisect import bisect_left, bisect_right
input = sys.stdin.readline
sys.setrecursionlimit(10 ** 7)

def sol():
    N, x = map(int, input().split())

    if x == 1 or x == 2 * N - 1:
        print('No')
    else:
        print('Yes')
        ans = [None] * (2 * N - 1)
        numList = list(range(1, 2 * N))

        for i in range(3):
            ans[N - 2 + i] = x - 1 + i
            numList[x - 2 + i] = None
        if x > N:
            ans[N + 1] = 1
            numList[0] = None
        elif x < N:
            ans[N - 3] = 2 * N - 1
            numList[2 * N - 2] = None

        numList = list(filter(lambda n: n != None, numList))
        now = 0
        for a in ans:
            if a == None:
                print(numList[now])
                now += 1
            else:
                print(a)



sol()