Digits in optimal order

Given two natural numbers $m$ and $n$ , you must construct a number $x$ using the digits $\{1, \dots, n\}$ (exactly one of each) such that no (non-empty) prefix of $x$ is a multiple of $m$ . For example, with $m=3$ and $n=4$ , $x=2314$ is not a valid order, because $231$ is a multiple of $3$ . In contrast, $x=4312$ is a valid order, because neither $4$ , nor $43$ , nor $431$ , nor $4312$ are multiples of $3$ .

Moreover, you have a matrix $M[1..n][1..n]$ such that $M[i][j]$ contains the prize that is obtained if digit $j$ is placed immediately to the right of digit $i$ . Maximize the total sum of the prizes.

Input

The input consists of several cases. Each case starts with $m$ and $n$ , followed by $M$ : $n$ rows, each with $n$ natural numbers between 1 and $10^6$ , except for the diagonal, that is filled with zeros. You can assume that $3 \le m \le 1000$ and $2 \le n \le 9$ .

Output

For each case, print the maximum possible prize. If no $x$ can be constructed, print 0.

Problem information

Author: Unknown
Translator: Albert Oliveras

Generation: 2026-01-25T11:22:09.366Z