
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $n$  be a positive integer. Assume that  $n$  numbers are to be chosen from the table $\begin{array}{cccc}0 & 1 & \cdots & n-1 n & n+1 & \cdots & 2n-1 \vdots & \vdots & \ddots & \vdots(n-1)n & (n-1)n+1 & \cdots & n^2-1\end{array} $ with no two of them from the same row or the same column. Find the maximal value of the product of these  $n$  numbers.

Present the answer in LaTex format: \boxed{Your answer}
user