
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
Given an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with one of its foci coinciding with the focus of the parabola $y^2 = 4\sqrt{3}x$, and the point $(\sqrt{3}, \frac{1}{2})$ lies on the ellipse $C$.
(1) Find the equation of the ellipse $C$;
(2) Let line $l$: $y = kx + m (k \neq 0)$ intersect the ellipse $C$ at points $P$ and $Q$, and a rhombus with $PQ$ as one of its diagonals has one vertex at $(-1, 0)$. Find the maximum area of triangle $\triangle OPQ$ ($O$ is the origin of the coordinate system).

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