
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) with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ and a focal length of $2\sqrt{3}$, and the ellipse (C) passes through point $A(1, \frac{\sqrt{3}}{2})$,
(I) Find the equation of ellipse (C);
(II) If $O$ is the coordinate origin, the line $l: y = kx + m$ that does not pass through the origin intersects the ellipse at two distinct points $P(x_1, y_1)$ and $Q(x_2, y_2)$, and $y_1y_2 = k^2x_1x_2$, find the slope $k$ of line $l$;
(III) Under the conditions of (II), find the maximum area of $\triangle OPQ$.

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