CF2053F Earnest Matrix Complement

3, 2, 1, ... We are the — RiOI Team! — Felix & 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 $$\sum_{u=1}^k \sum_{i=1}^{n-1} c_{u,i} \cdot c_{u,i+1}. $$ You have to find the maximum possible beauty of $ A$$$ after filling in the blanks optimally.

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 $ .

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

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