
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 \( n \geq 2 \) be an integer. Consider an \( n \times n \) chessboard divided into \( n^2 \) unit squares. We call a configuration of \( n \) rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer \( k \) such that for every happy configuration of rooks, we can find a \( k \times k \) square without a rook on any of its \( k^2 \) unit squares.

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