
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 $A(0,-2)$, the ellipse $E$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ has an eccentricity of $\frac{\sqrt{3}}{2}$, $F$ is the right focus of the ellipse $E$, the slope of the line $AF$ is $\frac{\sqrt{6}}{3}$, and $O$ is the origin.<br/>$(1)$ Find the equation of the ellipse;<br/>$(2)$ Let the moving line $l$ passing through point $A$ intersect the ellipse $E$ at points $P$ and $Q$. When the area of $\triangle OPQ$ is maximized, find the equation of the line $l$.

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