
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 $PQR,$ $PQ = 24,$ $QR = 25,$ $PR = 7,$ and point $H$ is the intersection of the altitudes (the orthocenter). 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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