
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  $ABC,$   $AB=13,$   $BC=14,$   $AC=15,$  and point  $G$  is the intersection of the medians. Points  $A',$   $B',$  and  $C',$  are the images of  $A,$   $B,$  and  $C,$  respectively, after a  $180^\circ$  rotation about  $G.$  What is the area if the union of the two regions enclosed by the triangles  $ABC$  and  $A'B'C'?$ 

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