
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 \( DEF \) where \( DE = 5, EF = 12, DF = 13 \), and point \( H \) is the centroid. After rotating triangle \( DEF \) by \( 180^\circ \) around \( H \), vertices \( D', E', F' \) are formed. Calculate the area of the union of triangles \( DEF \) and \( D'E'F' \).

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