
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 $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $e = \frac{\sqrt{3}}{2}$, and the point $(1, \frac{\sqrt{3}}{2})$ lies on the ellipse $E$.
(I) Find the equation of the ellipse $E$;
(II) Line $l$ intersects ellipse $E$ at points $A$ and $B$, and the perpendicular bisector of segment $AB$ passes through point $(0, \frac{1}{2})$. Find the maximum area of $\triangle AOB$ ($O$ is the origin).

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