
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  $AB=13$ ,  $BC=14$ ,  $CA=15$ . Let  $G_A$  be the centroid of triangle  $HBC$ , and define  $G_B$ ,  $G_C$  similarly. Determine the area of triangle  $G_AG_BG_C$ .

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