
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
On a $4 \times 4$ checkered board, Petya colors several cells. Vasya wins if he is able to cover all these cells with non-overlapping L-shaped tiles, each consisting of three cells, without any part of the tiles extending beyond the border of the square. What is the minimum number of cells Petya needs to color so that Vasya cannot win?

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