
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
Let the intersection point of the parabola $y^2=8x$ be $F$, and a fixed line $l: x=4$. Let $P$ be a moving point on the plane. A perpendicular line is drawn from point $P$ to line $l$, and the foot of the perpendicular is $Q$. Given that $(\overrightarrow{PQ} + \sqrt{2} \overrightarrow{PF}) \cdot (\overrightarrow{PQ} - \sqrt{2} \overrightarrow{PF}) = 0$,

(1) Find the equation of the trajectory $C$ of point $P$;
(2) The line $l$ is any tangent line of the circle $O: x^2+y^2=r^2$, and $l$ intersects curve $C$ at points $A$ and $B$. If the circle with diameter $AB$ always passes through the origin, find the equation of circle $O$ and the range of values for $|AB|$.

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