
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
Let \( ABC \) be a triangle with orthocenter \( H \); suppose that \( AB = 13 \), \( BC = 14 \), \( CA = 15 \). Let \( G_A \) be the centroid of triangle \( HBC \), and define \( G_B \), \( G_C \) similarly. Determine the area of triangle \( G_A G_B G_C \).

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