
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: $$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \quad (a > b > 0)$$ has one focus at $F(\sqrt{3}, 0)$, and the distance from the major vertex to the point A(0, -2) is $2\sqrt{2}$, with O being the origin of the coordinate system.
(1) Determine the equation of the ellipse C.
(2) Let the moving line l pass through point A and intersect ellipse C at points M and N. When the area of triangle OMN is maximized, find the equation of l.

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