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 <cstdio>
#define N 200005
using namespace std;

int n, x, a[N];
bool v[N];

void f(int p, int q) {
	a[n + p] = x + q;
	v[x + q] = 1;
}

int main()
{
	int i, j;
	cin >> n >> x;
	if (x == 1 || x == n * 2 - 1) {cout << "No"; return 0;}
	cout << "Yes\n";
	if (x != 2) f(-1, -1), f(0, 0), f(1, 1), f(2, -2);
	else f(-1, 1), f(0, 0), f(1, -1), f(2, 2);
	for (i = j = 1; i < 2 * n; i++) {
		if (!a[i]) {
			for (; v[j]; j++);
			a[i] = j++;
		}
		printf("%d\n", a[i]);
	}
    return 0;
}