
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 line $l$ passing through point $P(0, 2)$ intersects the parabola $C: y^2 = 4x$ at points $A$ and $B$, with $O$ being the origin of the coordinate system.  
(1) If the circle with diameter $AB$ passes through the origin $O$, find the equation of line $l$;  
(2) If the perpendicular bisector of segment $AB$ intersects the x-axis at point $Q$, find the range of the area of $\triangle POQ$.

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