
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $P(1, \frac{\sqrt{3}}{2})$, and its eccentricity $e = \frac{\sqrt{3}}{2}$.  
(I) Find the standard equation of the ellipse $C$;  
(II) Suppose a line $l$ passing through point $E(0, -2)$ intersects $C$ at points $P$ and $Q$, find the maximum value of the area of $\triangle OPQ$.

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