
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 $P(0,-2)$, and an ellipse $E$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ $(a > b > 0)$ with eccentricity $\dfrac{\sqrt{2}}{2}$, let $F$ be the right focus of ellipse $E$, and the slope of line $PF$ is $2$, with $O$ being the origin.  
$(1)$ Find the equation of ellipse $E$;  
$(2)$ A line $l$ intersects the circle $O$: $x^{2}+y^{2}=3$ and forms a chord of length $3$, and intersects ellipse $E$ at points $A$ and $B$. Find the maximum area of $\triangle AOB$.

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