
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
Let  $ n$  be a positive integer. Consider an  $ n\times n$  matrix with entries  $ 1,2,...,n^2$  written in order, starting at the top left and moving along each row in turn left-to-right. (e.g. for  $ n \equal{} 3$  we get  $ \left[\begin{array}{ccc}1 & 2 & 3 
4 & 5 & 6 
7 & 8 & 9\end{array}\right]$ )
We choose  $ n$  entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries?

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