
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, and then Vasya places an L-shaped piece covering three cells on the board and tells Petya which of the marked cells are covered. Vasya wins if Petya cannot uniquely determine the position of the L-shaped piece. For what is the smallest \( k \) such that Petya can mark the cells in a way that Vasya cannot win?

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