
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
A moving circle passes through a fixed point R(0, 2) and intersects the x-axis at a segment MN with a length of 4. The line $l: y = kx + t$ ($t > 0$) intersects the y-axis at point Q.  
(1) Find the equation of the trajectory E of the center of the moving circle;  
(2) The line $l$ intersects the trajectory E at points A and B. Tangents to the trajectory E at points A and B intersect at point P. If $\tan \angle APB = \frac{|\vec{PQ}| \cdot |\vec{AB}|}{\vec{PA} \cdot \vec{PB}}$, determine whether point Q is a fixed point. If it is, find the coordinates of point Q; if not, explain why.

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