
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 $CD=4.$ Suppose that $\triangle ABD$ is right-angled at $B$ and that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of a right triangle. Find the maximum possible value of the area of $ABCD$.
A) $\frac{1}{2}xy + 6$
B) $27$
C) $16\sqrt3$
D) $9 + 12\sqrt3$
E) $30$

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