
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $e = \frac{\sqrt{3}}{2}$, let A and B be the endpoints of the major and minor axes respectively, M be the midpoint of AB, and O be the origin with $|\vec{OM}| = \frac{\sqrt{5}}{2}$.
(I) Find the equation of the ellipse.
(II) The line l passing through $(-1, 0)$ intersects the ellipse at points P and Q. Find the equation of the line l when the area of $\triangle POQ$ is at its maximum.

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