
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 play the following game. Petya marks $k$ cells on a $9 \times 9$ board, after which Vasya places an L-shaped piece consisting of three cells on the board and tells Petya which of the marked cells are covered by the piece. Vasya wins if Petya cannot uniquely determine the position of the L-shaped piece. 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}
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