
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 $PQR$, $PQ = 12$, $QR = 15$, $PR = 13$, and point $H$ is the intersection of the medians. Points $P'$, $Q'$, and $R'$, are the images of $P$, $Q$, and $R$, respectively, after a $180^\circ$ rotation about $H$. What is the area of the union of the two regions enclosed by the triangles $PQR$ and $P'Q'R'$?

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