
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
Given a regular triangular pyramid \( S-ABC \) with height \( SO = 3 \) and the side length of the base is 6. A perpendicular is drawn from point \( A \) to the opposite face \( SBC \), with foot \( O' \). On \( AO' \), a point \( P \) is chosen such that \( \frac{AP}{PO'} = 8 \). Find the area of the cross-section through point \( P \) that is parallel to the base.

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