
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 $A(0,-2)$, an ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with its long axis length to short axis length ratio equal to $2$. $F$ is the right focus of the ellipse, and the slope of line $AF$ is $\frac{2 \sqrt{3}}{3}$. $O$ is the coordinate origin.

(I) Find the equation of $E$;

(II) Let the moving straight line $l$ pass through point $A$ and intersect with $E$ at points $P, Q$. When the area of $\Delta POQ$ is maximized, find the equation of $l$.

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