
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 $13 \times 13$ board, after which Vasya places a $1 \times 6$ rectangle on the board and informs Petya which of the marked cells he has covered (the rectangle can be rotated). Vasya wins if Petya cannot unambiguously determine the position of the rectangle. What is the smallest $k$ for which Petya can mark the cells in such a way that Vasya cannot win?

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