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 <iostream>
#include <algorithm>

using namespace std;

int main() {
    int n, x;
    cin >> n >> x;
    if (n == x) {
        cout << "Yes" << '\n';
        for (int i=1; i<2*n; i++) cout << i << '\n';
    } else if (1 < x && x < 2*n - 1) {
        cout << "Yes" << '\n';
        int i=2;
        for ( ; i<n; i++) 
            cout << i + (i >= x) << '\n';
        cout << 1 << '\n' 
             << x << '\n'
             << 2*n - 1 << '\n';
        for ( ; i+2<2*n; i++) 
            cout << i + (i >= x) << '\n';
    } else
        cout << "No" << '\n';
    return 0;
}