
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
Petya and Vasya are playing the following game. Petya marks \( k \) cells on a \( 9 \times 9 \) board, after which Vasya places a corner consisting of three cells on the board and informs Petya which of the marked cells he has covered. Vasya wins if Petya cannot uniquely determine the position of the corner. What is the smallest \( k \) for which Petya can mark the cells so that Vasya cannot win?

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