
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 $S$ with its vertex at the origin and its focus on the x-axis, the three vertices of $\triangle ABC$ are all on the parabola, and the centroid of $\triangle ABC$ is the focus of the parabola. If the equation of the line $l$ containing $BC$ is $4x+y-20=0$.

(I) Find the equation of the parabola $S$;  
(II) Let $O$ be the origin, and $P$, $Q$ be two moving points on the parabola $S$, satisfying $PO \perp OQ$. Try to explain whether the moving line $PQ$ passes through a fixed point.

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