
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 first \( n^{2} \) positive integers \( 1, 2, 3, \cdots, n^{2} \) in the following table:
\[ 
\begin{array}{ccccc}
1 & 2 & 3 & \cdots & n \\
n+1 & n+2 & n+3 & \cdots & 2n \\
\vdots & \vdots & \vdots & & \vdots \\
(n-1)n+1 & (n-1)n+2 & (n-1)n+3 & \cdots & n^{2}
\end{array}
\]

Now, if a number is randomly selected from the table, then the entire row and column containing this number are deleted. This process is repeated for the remaining \( (n-1)^{2} \) numbers, for a total of \( n \) steps. What is the sum of the \( n \) selected numbers?

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