Problem Description

Given positive integers $N$ and $K$.

An integer sequence of length $NK$ where each integer between $1$ and $N$ appears exactly $K$ times is called a good sequence.

Let $S$ be the number of good sequences. Find the $\operatorname{floor}((S+1)/2)$-th good sequence in lexicographical order. Here, $\operatorname{floor}(x)$ represents the largest integer not exceeding $x$.

What is lexicographical order for sequences? A sequence $S = (S_1, S_2, \ldots, S_{|S|})$ is smaller in lexicographical order than a sequence $T = (T_1, T_2, \ldots, T_{|T|})$ if either of the following holds:

1. $ |S| < |T| $ and $(S_1, S_2, \ldots, S_{|S|}) = (T_1, T_2, \ldots, T_{|S|})$.
2. There exists an integer $1 \leq i \leq \min\{|S|, |T|\}$ such that both of the following are true:
   - $(S_1, S_2, \ldots, S_{i-1}) = (T_1, T_2, \ldots, T_{i-1})$
   - $ S_i $ is smaller than $ T_i $ (as a number).

Input Format

The input is given from the standard input in the following format:

$ N $ $ K $

Output Format

Output the resulting sequence with elements separated by spaces.

Sample Input and Output

Sample Input #1

2 2

Sample Output #1

1 2 2 1

Sample Input #2

1 5

Sample Output #2

1 1 1 1 1

Sample Input #3

6 1

Sample Output #3

3 6 5 4 2 1

Sample Input #4

3 3

Sample Output #4

2 2 2 1 3 3 3 1 1

Notes/Hints

Constraints

• $1 \leq N \leq 500$
• $1 \leq K \leq 500$
• All values in the input are integers.

Sample Explanation 1

The good sequences are as follows:
- $ (1,1,2,2) $
- $ (1,2,1,2) $
- $ (1,2,2,1) $
- $ (2,1,1,2) $
- $ (2,1,2,1) $
- $ (2,2,1,1) $

Thus, the $3$-rd sequence in lexicographical order, $ (1,2,2,1) $, is the answer.

见题面 Input 部分

见题面 Output 部分