
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 situated in the Cartesian plane with $A$ at the origin $(0, 0)$, $B$ on the x-axis at $(x, 0)$ where $x > 0$, $BC = 3$, and $CD = 5$. Assume that the centroids of $\triangle ABC, \triangle BCD,$ and $\triangle ACD$ form an isosceles triangle with the centroid of $\triangle ABC$ as the vertex angle. Determine the maximum possible value of the area of $ABCD$.
A) $15$
B) $12$
C) $10$
D) $18$
E) $9$

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