
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
Arrange the natural numbers $1, 2, \cdots, k^{2}$ in a square matrix as follows:
\begin{tabular}{llllc}
1 & 2 & 3 & $\cdots$ & $k$ \\
$k+1$ & $k+2$ & $k+3$ & $\cdots$ & $2 k$ \\
$\vdots$ & $\vdots$ & $\vdots$ & & $\vdots$ \\
$(k-1) k+1$ & $(k-1) k+2$ & $(k-1) k+3$ & $\cdots$ & $k^{2}$
\end{tabular}

From this table, select any number, then delete all numbers in the same row and column as the selected number. Next, select another number from the remaining $(k-1)^{2}$ numbers and delete all numbers in the same row and column as the selected number. Continue this process until $k$ different numbers have been selected, forming a $k$-element subset of $\{1, 2, \cdots, k^{2}\}$. An interesting question is: How many different $k$-element subsets of $\{1, 2, \cdots, k^{2}\}$ can be obtained using this method? What is the sum of the $k$ numbers in each of these subsets?

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