
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
It is known that a moving circle P passes through point A(2, 0) and cuts a chord of length 4 on the y-axis.

(1) Find the equation of the trajectory C of the center P of the moving circle;
(2) Suppose A$(x_1, y_1)$ and B$(x_2, y_2)$ are two moving points on 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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