
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 \( PQR \), \( PQ = 15 \), \( QR = 17 \), \( PR = 8 \), and point \( M \) is the centroid. Points \( P' \), \( Q' \), and \( R' \) are the images of \( P \), \( Q \), and \( R \), respectively, after a \( 180^\circ \) rotation about \( M \). 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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