
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 one of its foci at $F(2,0)$ and its eccentricity as $\frac{\sqrt{6}}{3}$. A line $l$ passing through the focus $F$ intersects the ellipse $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $D$, and $O$ being the origin. The line passing through $O$ and $D$ intersects the ellipse at points $M$ and $N$.
1. Find the equation of the ellipse $C$;
2. Find the maximum area of the quadrilateral $AMBN$.

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