
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  $n \ge 2$  be an integer. Consider an  $n \times n$  chessboard consisting of  $n^2$  unit squares. A configuration of  $n$  rooks on this board is *peaceful* if every row and every column contains exactly one rook. Find the greatest positive integer  $k$  such that, for each peaceful configuration of  $n$  rooks, there is a  $k \times k$  square which does not contain a rook on any of its  $k^2$  unit squares.

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