
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{\sqrt{6}}{3}$, and the ellipse $C$ passes through the point $({\sqrt{3},1})$ and intersects with the line $l$ passing through the right focus $F$ of the ellipse $C$ at points $A$ and $B$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let $O$ be the origin, find the maximum area of $\triangle OAB$ and the equation of the line $l$ at that time.

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