
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
In triangle \(ABC\) on side \(AB\) points \(E\) and \(F\) lie. The area of triangle \(AEC\) is \(1 \text{ cm}^2\), the area of triangle \(EFC\) is \(3 \text{ cm}^2\), and the area of triangle \(FBC\) is \(2 \text{ cm}^2\). Point \(T\) is the centroid of triangle \(AFC\), and point \(G\) is the intersection of lines \(CT\) and \(AB\). Point \(R\) is the centroid of triangle \(EBC\), and point \(H\) is the intersection of lines \(CR\) and \(AB\).

Determine the area of triangle \(GHC\).

(E. Semerádová)

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