
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 an ellipse $C$ with foci at $(-2,0)$ and $(2,0)$, and it passes through $P(2, \frac{\sqrt{6}}{3})$.

$(1)$ Find the standard equation of ellipse $C$;  
$(2)$ A line $l$ passes through the right focus $F$ of ellipse $C$ and intersects the ellipse at points $A$ and $B$. When the area of $\triangle OAB$ is maximized, find the equation of line $l$.

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