
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 $BC=3$ and $CD=4$. Assume the centroids of $\triangle ABC, \triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. Determining the side lengths AB and AD being equal, what is the maximum possible area of $ABCD$?
A) $\frac{25\sqrt{3}}{4} + 6$
B) $\frac{28\sqrt{3}}{4} + 4$
C) $\frac{32\sqrt{3}}{4} + 2$
D) $\frac{37\sqrt{3}}{4} + 3$

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