
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 intersection of the medians. Points $D',$ $E',$ and $F',$ are the images of $D,$ $E,$ and $F,$ respectively, after a $180^\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles $DEF$ and $D'E'F'?$

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