
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
On a \(50 \times 50\) checkerboard, Petya places 50 rooks such that no two rooks attack each other, and Vasya chooses a \(k \times k (k \leq 50)\) square on the board. For what values of \(k\) will Vasya always be able to choose a square that contains no rooks, regardless of Petya's actions?

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