
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

Mouse Jerry decided to give a birthday pie to cat Tom in the shape of an $8 \times 8$ square. He filled three pieces marked with the letter "P" with fish, two pieces marked with the letter "K" with sausage, and one more piece contains both fish and sausage, but this piece is not marked (all other pieces are without filling). Additionally, Jerry informed Tom that in any $6 \times 6$ square, there are at least 2 pieces with fish, while in any $3 \times 3$ square, there is no more than one piece with sausage.

What is the minimum number of pieces that Tom needs to eat to ensure he gets a piece with both fish and sausage?

|  |  |  |  |  |  |  |  |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
|  | P |  |  |  |  |  |  |
|  | K |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  |  |  |  |  |
|  |  |  |  | P | K |  |  |
|  |  |  |  |  |  |  |  |
|  | P |  |  |  |  |  |  |

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