
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 $1, 2, \cdots, k^{2}$ into a square number table. Then, select one number from the table and delete the row and column containing that number. Repeat this process for the remaining $(k-1)^{2}$ numbers in the square table, continuing in the same manner until there have been $k$ selections. Find the sum of the selected $k$ numbers.

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