
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 with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ and an eccentricity of $\frac{\sqrt{6}}{3}$, the length of the major axis is $2\sqrt{3}$. Line $l$: $y = kx + m$ intersects the ellipse at two distinct points $A$ and $B$.

(I) Find the equation of the ellipse;

(II) If $m = 1$, and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$ (where $O$ is the origin), find the value of $k$;

(III) If the distance from the origin $O$ to line $l$ is $\frac{\sqrt{3}}{2}$, find the maximum area of $\triangle AOB$.

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