
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 equation of circle $C_{1}$ is $(x-1)^{2}+y^{2}=9$, and the equation of circle $C_{2}$ is $(x+1)^{2}+y^{2}=1$. The moving circle $C$ is internally tangent to circle $C_{1}$ and externally tangent to circle $C_{2}$.  
$(1)$ Find the equation of the trajectory $E$ of the center $C$ of the moving circle;  
$(2)$ Given two fixed points $P(-2,0)$ and $Q(2,0)$ in the plane, a line $l$ passing through the point $(1,0)$ intersects the trajectory $E$ 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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