
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{1}{2}$, and points $P(2,3)$ and $Q(2,-3)$ on the ellipse:

(I) Find the equation of ellipse $C$;

(II) Let $A$ and $B$ be points on the ellipse on opposite sides of the line $PQ$. If the slope of the line $AB$ is $\frac{1}{2}$, find the maximum area of the quadrilateral $APBQ$.

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