
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
Let $ABCD$ be a convex quadrilateral with sides $BC = 3$ and $CD = 5$. Suppose that the centroids of triangles $\triangle ABC$, $\triangle BCD$, and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of the area of $ABCD$?
A) $8.5 \sqrt{3}$
B) $8.5 \sqrt{3} + 7.5$
C) $8.5 + 7.5 \sqrt{3}$
D) $17 \sqrt{3}$

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