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 CPP):

Solution:
#include <bits/stdc++.h>

using namespace std;

int n, x;
int arr[200100];

int main() {
    int i;
    scanf("%d%d",&n,&x);
    if (x>1&&x<2*n-1) {
        printf("Yes\n");
        arr[n-1] = x;
        int p = 1;
        for (i=n;i<2*n-1;i++) {
            if (p==x) p++;
            arr[i] = p++;
        }
        p = 2*n-1;
        for (i=n-2;i>=0;i--) {
            if (p==x) p--;
            arr[i] = p--;
        }
        for (i=0;i<2*n-1;i++) printf("%d\n",arr[i]);
    }
    else printf("No\n");
    return 0;
}