
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
In the Cartesian coordinate system, the eccentricity of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $\frac{1}{2}$, and the point $M(1,\frac{3}{2})$ is on the ellipse $C$.

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

$(2)$ Given two fixed points in the plane $P(-2,0)$ and $Q(2,0)$, a line $l$ passing through the point $(1,0)$ intersects the ellipse $C$ at points $A$ and $B$. Find the maximum area of the quadrilateral $APBQ$.

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