
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 \( P-ABC \) with a volume of \( 9 \sqrt{3} \), the dihedral angle between the face \( PAB \) and the base \( ABC \) is \( 60^\circ \). Point \( D \) is on the line segment \( AB \) such that \( AD = \frac{1}{6} AB \), point \( E \) is on the line segment \( AC \) such that \( AE = \frac{1}{6} AC \), and point \( F \) is the midpoint of \( PC \). The plane \( DEF \) intersects the line segment \( PB \) at point \( G \). Find the area of the quadrilateral \( DEFG \).

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