Primonimo

Primonimo is a game played on an $n \times m$ board. Every square holds an integer between $1$ and $p$, where $p$ is a prime. One move picks a square, and every square in the same row or the same column as the picked square grows by $1$. The picked square belongs to both that row and that column, but it still grows by only $1$. A square that held $p$ goes back to $1$.

You win when every square shows $p$. Given the starting board, find a sequence of moves that wins the game.

Squares are numbered from $1$ to $nm$ in row-major order, so the square in row $i$ and column $j$ (both counted from 1) has number $(i-1)m + j$.

The input is one test case. The first line has three integers $n$, $m$, $p$: the number of rows $n$ ($1 \le n \le 20$), the number of columns $m$ ($1 \le m \le 20$), and a prime $p$ ($2 \le p \le 97$). Each of the next $n$ lines has $m$ integers between $1$ and $p$.

If no winning sequence exists, print $-1$.

Otherwise the answer is fixed by this rule. The order of the moves does not change the result, so only the number of times each square is picked matters. Let $x_t$ be the number of times square $t$ is picked, and consider only the count vectors with $0 \le x_t \le p-1$. When several count vectors win, take the lexicographically smallest $(x_1, x_2, \ldots, x_{nm})$: the smallest $x_1$, then among those the smallest $x_2$, and so on.

Print $k = x_1 + \cdots + x_{nm}$ on the first line. On the second line print the $k$ chosen square numbers from the smallest number up, writing number $t$ exactly $x_t$ times. When $k = 0$ the second line is empty. This $k$ is always at most $p \cdot n \cdot m$.