
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 a point $A(0,-2)$, an ellipse $E: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{3}}{2}$. Let $F$ be the right focus of the ellipse $E$, and the slope of the line $AF$ is $\frac{2\sqrt{3}}{3}$. $O$ is the origin.

(1) Find the equation of $E$;

(2) Let a line $l$ pass through point $A$ and intersect the ellipse $E$ at points $P$ and $Q$. When the area of $\triangle OPQ$ is maximized, find the equation of line $l$.

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