
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
The numbers \(1, 2, \ldots, k^{2}\) are arranged in a square table:

$$
\begin{array}{cccc}
1, & 2, & \ldots, & k \\
k+1, & k+2, & \ldots, & 2k \\
\ldots & \ldots & \ldots & \ldots \\
(k-1)k+1, & \ldots, & \ldots, & k^{2} .
\end{array}
$$

We randomly select a number from this table, then cross out the row and column containing this number. We continue this process with the remaining \((k-1)^{2}\) numbers, and so on, for \(k\) steps. Find the sum of the selected numbers.

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