
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 point \( P \) inside an equilateral triangle \( ABC \), perpendiculars are drawn from \( P \) to the three sides, with feet of the perpendiculars denoted as \( D, E, F \) respectively. Lines \( AP, BP, \) and \( CP \) are connected. If the area of the equilateral triangle \( ABC \) is 2028 square centimeters and the areas of triangles \( PAD \) and \( PBE \) are each 192 square centimeters, then the area of triangle \( PCF \) is \( \quad \) square centimeters.

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