
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 $1 \times 4$ rectangle on the board and tells Petya which of the marked cells are covered by the rectangle (the rectangle can be rotated). Vasya wins if Petya cannot uniquely determine the position of the rectangle. 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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