
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 $DEF,$ $DE = 8,$ $EF = 17,$ $DF = 15,$ and point $H$ is the centroid. Triangle $D'E'F'$ is the image of triangle $DEF$ after reflecting across the line passing through $H$ and vertex $D.$ Calculate the area of the union of the regions enclosed by triangles $DEF$ and $D'E'F'.$

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