
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 $C$ with its center at the origin and its foci on the $x$-axis, and its eccentricity equal to $\frac{1}{2}$. One of its vertices is exactly the focus of the parabola $x^{2}=8\sqrt{3}y$.

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

(Ⅱ) If the line $x=-2$ intersects the ellipse at points $P$ and $Q$, and $A$, $B$ are points on the ellipse located on either side of the line $x=-2$.

(i) If the slope of line $AB$ is $\frac{1}{2}$, find the maximum area of the quadrilateral $APBQ$;

(ii) When the points $A$, $B$ satisfy $\angle APQ = \angle BPQ$, does the slope of line $AB$ have a fixed value? Please explain your reasoning.

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