
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 base side length of 6. A perpendicular line is drawn from point \( A \) to the corresponding lateral face \( SBC \), with the foot of the perpendicular being \( D \). A point \( P \) is taken on \( AD \), such that \( \frac{AP}{PD} = 8 \). Find the area of the cross-section that passes through point \( P \) and is parallel to the base.

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