
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
The side of the base of a regular quadrilateral pyramid \( \operatorname{ABCDP} \) (with \( P \) as the apex) is \( 4 \sqrt{2} \), and the angle between adjacent lateral faces is \( 120^{\circ} \). Find the area of the cross-section of the pyramid by a plane passing through the diagonal \( BD \) of the base and parallel to the lateral edge \( CP \).

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