
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 point A (0, -2), and an ellipse E described by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$) has an eccentricity of $\frac{\sqrt{3}}{2}$. F is the right focus of the ellipse E, and the slope of line AF is $\frac{2\sqrt{3}}{3}$. O is the origin of the coordinate system.
(1) Find the equation of the ellipse E.
(2) Let line l pass through point A and intersect the ellipse E at points P and Q. When the area of triangle OPQ is the maximum, find the equation of line l.

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