
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 $AB=4$ and $AD=3.$ Suppose that the centroids of $\triangle ABD$, $\triangle BCD$, and $\triangle ABC$ form the vertices of a right triangle. Determine the maximum possible value of the area of $ABCD$.
A) \(6\)
B) \(7.5\)
C) \(9\)
D) \(10.5\)

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