
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
In the Cartesian coordinate system $xOy$, let $C$ be the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ with its left focus as $F_1$, and the two endpoints of the minor axis as $A$ and $B$, with $\angle AF_1B=60^{\circ}$. Given that the point $(\sqrt{3}, \frac{1}{2})$ lies on $C$.
(I) Find the equation of the ellipse $C$;
(II) If the line $l: y=kx+m (k>0)$ is tangent to both the ellipse $C$ and the circle $O$ at points $P$ and $Q$ respectively, find the equation of the line $l$ when the area of $\triangle OPQ$ is maximized.

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