Score : $1000$ points

Problem Statement

There are positive integers $N, M$, and an $N \times M$ positive integer matrix $A_{i, j}$. For two (strictly) increasing positive integer sequences $X = (X_1, \ldots, X_N)$ and $Y = (Y_1, \ldots, Y_M)$, we define the penalty $D(X, Y)$ as $\max_{1 \leq i \leq N, 1 \leq j \leq M} |X_i Y_j - A_{i, j}|$.

Find two (strictly) increasing positive integer sequences $X$ and $Y$ such that $D(X, Y)$ is the smallest possible.

Constraints

• $1 \leq N, M \leq 5$
• $1 \leq A_{i, j} \leq 10^9$ ($1 \leq i \leq N$, $1 \leq j \leq M$)
• All values in the input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$

$A_{1,1}$ $\ldots$ $A_{1,M}$

$\vdots$

$A_{N,1}$ $\ldots$ $A_{N,M}$

Output

Output your answer in the following format:

$D_{min}$

$X_1$ $\ldots$ $X_N$

$Y_1$ $\ldots$ $Y_M$

Here, $D_{min}$ is the smallest possible penalty and the following conditions should be satisfied:

• $D(X, Y)$ should be equal to $D_{min}$.
• $X_i < X_{i + 1}$ ($1 \leq i \leq N - 1$).
• $Y_j < Y_{j + 1}$ ($1 \leq j \leq M - 1$).
• $1 \leq X_i \leq 2\cdot 10^9$ ($1 \leq i \leq N$).
• $1 \leq Y_j \leq 2\cdot 10^9$ ($1 \leq j \leq M$).

One can show that there is an optimal solution satisfying the last two conditions.

1 1
853922530

0
31415
27182

3 3
4 4 4
4 4 4
4 4 4

5
1 2 3 
1 2 3

3 4
4674 7356 86312 100327
8737 11831 145034 167690
47432 66105 809393 936462

357
129 216 1208 
39 55 670 775