
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
We write the numbers $1, 2, \ldots, n^{2}$ sequentially into an $n \times n$ grid:

$$
\begin{array}{cccc}
1 & 2 & \ldots & n \\
n+1 & n+2 & \ldots & 2n \\
\vdots & \vdots & \ddots & \vdots \\
n^{2}-n+1 & n^{2}-n+2 & \ldots & n^{2}
\end{array}
$$

From each row, we select one number such that no two selected numbers are in the same column. What are the possible values of the sum of the selected numbers?

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