
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 that $A$ and $B$ are respectively the top and right vertices of the ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with an eccentricity of $\frac{\sqrt{3}}{2}$. The distance from the right focus $F_2$ to the line $AB$ is $\frac{2\sqrt{5} - \sqrt{15}}{5}$.  
$(1)$ Find the equation of the ellipse $E$;  
$(2)$ A line $l$ passing through $M(0,2)$ intersects the ellipse $E$ at points $P$ and $Q$, with $O$ being the origin. Find the maximum area of $\triangle OPQ$.

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