
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
Inside the equilateral triangle \(ABC\), there is a point \(P\). Perpendiculars are drawn from \(P\) to the three sides of the triangle, with the feet of these perpendiculars being \(D\), \(E\), and \(F\) respectively. Connect \(AP\), \(BP\), and \(CP\). 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 \(PCF\).

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