
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 composed of \( n^2 \) unit squares. If there is exactly one "rook" in each row and each column, the arrangement of the \( n \) rooks is called "peaceful." Find the largest positive integer \( k \) such that for any peaceful arrangement of \( n \) rooks, there exists a \( k \times k \) square with all \( k^2 \) unit squares unoccupied by rooks.

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