
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
Given natural numbers \( k \) and \( n \) such that \( 1 < k \leq n \). What is the smallest \( m \) for which the following statement holds: for any placement of \( m \) rooks on an \( n \times n \) board, it is always possible to select \( k \) rooks from these \( m \) such that no two of the selected rooks can attack each other?

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