King is a magician who can always accomplish incredible things using simple principles.

King has a stack of $n$ playing cards, all face down. King has assigned a unique number between $1$ and $n$ to each card. Initially, the number on the $i$-th card from the top is $p_i$. While performing a magic trick, King performs several operations such that the number on the $i$-th card from the top is exactly $i$.

To make the magic effect more deceptive, King only performs actions that appear ordinary. Omitting the specific sleight of hand, one operation by King can be simplified into the following steps:

1. Take a stack of cards from the top of the deck and place them on the table; this is called stack A (it can be empty).
2. Deal $m$ cards one by one from the top of the remaining deck onto the table, face down, to form stack B.
3. Place stack B back on top of the deck.
4. Place stack A back on top of the deck.

Please help King provide an operation sequence, or tell King that no such sequence exists.

Input

The first line contains three integers $T, n, m$, representing the test package ID, the number of cards in the deck, and the number of cards dealt in each operation.

The second line contains $n$ integers $p_1, p_2, \ldots, p_n$, representing the number on each card from top to bottom.

Output

If no operation sequence exists, output $-1$. Otherwise:

The first line contains a non-negative integer $k$, representing the number of operations.

The next $k$ lines each contain a non-negative integer $c_i\ (0 \le c_i \le n-m)$, representing the number of cards in stack A for the $i$-th operation.

Examples

Input 1

0 8 4
6 2 5 1 7 4 3 8

Output 1

4
0
2
3
1

Input 2

0 6 5
3 4 1 5 6 2

Output 2

-1

Subtasks

Each test package comes with parameters $lim_1, lim_2, \ldots, lim_5$. Let $S$ be the score of a test package. For the test cases within that package, the scoring is as follows:

• If the test case has no solution:
  • If the participant outputs a solution, they get $0$ points.
  • Otherwise, they get $S$ points.
• If the test case has a solution:
  • If the participant outputs that there is no solution or the provided sequence is invalid, they get $0$ points.
  • Otherwise, let $k$ be the number of operations in the participant's sequence; the score is: $$\frac{S}{5}\times \sum_{i=1}^{5}[k\le lim_i]$$

The score for the test package is the minimum score among all test cases within that package.

Constraints

For $100\%$ of the data, $1 \le m \le n \le 1000$, and $p$ is a permutation of $1$ to $n$.

For data where a solution exists, the data generation method is: after determining $n$ and $m$, $p$ is randomly generated from the set of permutations that have a solution.

The additional constraints for each test package are as follows:

For odd-numbered test packages, $m$ is guaranteed to be odd; for even-numbered test packages, $m$ is guaranteed to be even.