# Earnest Matrix Complement

## Problem Description

3, 2, 1, ... We are the — RiOI Team!

— Felix &amp; All, [Special Thanks 3](https://www.luogu.com.cn/problem/T351681)



- Peter: Good news: My problem T311013 is approved!
- $ \delta $ : I'm glad my computer had gone out of battery so that I wouldn't have participated in wyrqwq's round and gained a negative delta.
- Felix: \[thumbs\_up\] The problem statement concerning a removed song!
- Aquawave: Do I mourn my Chemistry?
- E.Space: ahh?
- Trine: Bread.
- Iris: So why am I always testing problems?

Time will pass, and we might meet again. Looking back at the past, everybody has lived the life they wanted.

Aquawave has a matrix $ A $ of size $ n\times m $ , whose elements can only be integers in the range $ [1, k] $ , inclusive. In the matrix, some cells are already filled with an integer, while the rest are currently not filled, denoted by $ -1 $ .

You are going to fill in all the unfilled places in $ A $ . After that, let $ c_{u,i} $ be the number of occurrences of element $ u $ in the $ i $ -th row. Aquawave defines the beauty of the matrix as

 $ $$$\sum_{u=1}^k \sum_{i=1}^{n-1} c_{u,i} \cdot c_{u,i+1}. $ $ </p><p>You have to find the maximum possible beauty of  $ A$$$ after filling in the blanks optimally.

## Input Format

The first line of input contains a single integer $ t $ ( $ 1 \leq t \leq 2\cdot 10^4 $ ) — the number of test cases. The description of test cases follows.

The first line of each test case contains three integers $ n $ , $ m $ , and $ k $ ( $ 2 \leq n \leq 2\cdot 10^5 $ , $ 2 \leq m \leq 2\cdot 10^5 $ , $ n \cdot m \leq 6\cdot 10^5 $ , $ 1 \leq k \leq n\cdot m $ ) — the number of rows and columns of the matrix $ A $ , and the range of the integers in the matrix, respectively.

Then $ n $ lines follow, the $ i $ -th line containing $ m $ integers $ A_{i,1},A_{i,2},\ldots,A_{i,m} $ ( $ 1 \leq A_{i,j} \leq k $ or $ A_{i,j} = -1 $ ) — the elements in $ A $ .

It is guaranteed that the sum of $ n\cdot m $ over all test cases does not exceed $ 6\cdot 10^5 $ .

## Output Format

For each test case, output a single integer — the maximum possible beauty.

## Sample #1

### Sample Input #1

```
9
3 3 3
1 2 2
3 1 3
3 2 1
2 3 3
-1 3 3
2 2 -1
3 3 6
-1 -1 1
1 2 -1
-1 -1 4
3 4 5
1 3 2 3
-1 -1 2 -1
3 1 5 1
5 3 8
5 -1 2
1 8 -1
-1 5 6
7 7 -1
4 4 4
6 6 5
-1 -1 5 -1 -1 -1
-1 -1 -1 -1 2 -1
-1 1 3 3 -1 -1
-1 1 -1 -1 -1 4
4 2 -1 -1 -1 4
-1 -1 1 2 -1 -1
6 6 4
-1 -1 -1 -1 1 -1
3 -1 2 2 4 -1
3 1 2 2 -1 -1
3 3 3 3 -1 2
-1 3 3 -1 1 3
3 -1 2 2 3 -1
5 5 3
1 1 3 -1 1
2 2 -1 -1 3
-1 -1 -1 2 -1
3 -1 -1 -1 2
-1 1 2 3 -1
6 2 7
-1 7
-1 6
7 -1
-1 -1
-1 -1
2 2
```

### Sample Output #1

```
4
4
10
10
8
102
93
58
13
```

## Hint

In the first test case, the matrix $ A $ is already determined. Its beauty is

 $ $$$\sum_{u=1}^k \sum_{i=1}^{n-1} c_{u,i} \cdot c_{u,i+1} = c_{1,1}\cdot c_{1,2} + c_{1,2}\cdot c_{1,3} + c_{2,1}\cdot c_{2,2} + c_{2,2}\cdot c_{2,3} + c_{3,1}\cdot c_{3,2} + c_{3,2}\cdot c_{3,3} = 1\cdot 1 + 1\cdot 1 + 2\cdot 0 + 0\cdot 1 + 0\cdot 2 + 2\cdot 1 = 4. $ $ </p><p>In the second test case, one can fill the matrix as follows:</p><p> $ $  \begin{bmatrix} 2 &3 &3 \\ 2 &2 &3 \end{bmatrix},  $ $ </p><p>and get the value  $ 4 $ . It can be proven this is the maximum possible answer one can get.</p><p>In the third test case, one of the possible optimal configurations is:</p><p> $ $  \begin{bmatrix} 1 &1 &1 \\ 1 &2 &1 \\ 1 &1 &4 \end{bmatrix}.  $ $ </p><p>In the fourth test case, one of the possible optimal configurations is:</p><p> $ $  \begin{bmatrix} 1 &3 &2 &3 \\ 1 &3 &2 &1 \\ 3 &1 &5 &1 \end{bmatrix}.  $ $ </p><p>In the fifth test case, one of the possible optimal configurations is:</p><p> $ $  \begin{bmatrix} 5 &5 &2 \\ 1 &8 &5 \\ 7 &5 &6 \\ 7 &7 &4 \\ 4 &4 &4 \end{bmatrix}.  $ $$$

## Limit



Time Limit
5.00s

Memory Limit
500.00MB

Tags: brute force, data structures, dp, greedy, implementation, math