
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
In each of the \( k \) cells of an \( n \times n \) square table, there is a one, and in the remaining cells, there are zeros. Find the maximum value of \( k \) for which, regardless of the initial positions of the ones, by swapping rows among themselves and columns among themselves, it is possible to achieve that all ones are above or on the secondary diagonal. (The secondary diagonal is the diagonal that runs from the bottom left corner to the top right corner.)

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