
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 an equilateral triangle \(ABC\) with an internal point \(P\). Perpendiculars are dropped from \(P\) to each of the three sides, landing on points \(D\), \(E\), and \(F\) respectively. Lines \(AP\), \(BP\), and \(CP\) are drawn. If the area of the equilateral triangle \(ABC\) is 2028 square centimeters, and the areas of triangles \(PAD\) and \(PBE\) are both 192 square centimeters, find the area of triangle \(PCF\) in square centimeters.

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