
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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{6}}{3}$ and a major axis length of $2\sqrt{3}$, and a line $l: y = kx + m$ that intersects the ellipse at two distinct points A and B.
(I) Find the equation of the ellipse.
(II) When line $l$ passes through the point $(0, 2)$, find the maximum area of triangle $AOB$.

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