
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 an eccentricity of $\frac{1}{2}$, one of its major axis endpoints is exactly the focus of the parabola $y^2=16x$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Given points $P(2,3)$ and $Q(2,-3)$ on the ellipse, let $A$ and $B$ be points on the ellipse located on opposite sides of line $PQ$.

   $(i)$ If the slope of line $AB$ is $\frac{1}{2}$, find the maximum area of quadrilateral $APBQ$.
   
   $(ii)$ As $A$ and $B$ move, satisfying $\angle APQ = \angle BPQ$, does the slope of line $AB$ remain constant? Please explain your reasoning.

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