
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 parabola and a point M outside it, two tangent lines can be drawn from M to the parabola. The line segment connecting the two tangent points is called the chord of tangent points corresponding to point M. It is known that the parabola is $x^2=4y$, and points P and Q are on the line $l: y=-1$. The chord of tangent points corresponding to points P and Q are AB and CD, respectively.
(1) When point P moves on line l, does line AB pass through a fixed point? If so, find the coordinates of this point; if not, explain the reason.
(2) When AB is perpendicular to CD, what are the positions of points P and Q when $|PQ|$ reaches its minimum value?

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