
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 a moving circle $P$ that passes through point $A(2,0)$, and the chord it intercepts on the $y$-axis has a length of $4$.
$(1)$ Find the equation of the trajectory $C$ of the center of the moving circle $P$;
$(2)$ Let $A(x\_1,y\_1)$, $B(x\_2,y\_2)$ be two points on the curve $C$, where $x\_1 \neq x\_2$, and $x\_1+x\_2=4$. The perpendicular bisector of segment $AB$ intersects the $x$-axis at point $Q$. Find the maximum area of $\triangle ABQ$.

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