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

$(1)$ Find the equation of $E$;

$(2)$ Suppose a variable line $l$ passing through point $A$ intersects $E$ at points $P$ and $Q$. When the area of $\triangle OPQ$ is maximized, find the equation of $l$.

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