Permutation

Limits: 2 sec., 256 MiB

This statement was translated automatically from Ukrainian.

Now that Zenik has guessed his password for Algotester on the seventy-fourth try, he can solve a simple problem before bed, even from his phone.

Problem: find any permutation \((p_1, p_2, \ldots, p_n)\) of the sequence \((1, 2, \ldots, n)\) such that \[\sum_{i=1}^n |p_i - i|=|p_1-1|+|p_2-2|+\dots+|p_n-n|=k,\] or state that no such permutation exists.

Input

The single line contains two integers \(n\) and \(k\).

Output

In the first line, print YES or NO — whether such a permutation exists or not.

If such a permutation exists, then in the second line print \(n\) integers \(p_i\) — the elements of the permutation that satisfy the condition.

Constraints

\(1 \le n \le 10^5\),

\(0 \le k \le 10^9\).

5 points: \(n \le 10\);

20 points: no additional constraints.

Samples

Input (stdin) сopy	Output (stdout) сopy
4 4	YES 1 4 2 3

Notes

Permutation — an ordered set without repetitions of the numbers \(1, 2, \ldots, n\).

In the example \(|p_1-1|+|p_2-2|+|p_3-3|+|p_4-4|=|1-1|+|4-2|+|2-3|+|3-4|=0+2+1+1=4\).