
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 the line $l: y=k(x-n)$ intersects the parabola $y^2=4x$ at points $A(x_1, y_1)$ and $B(x_2, y_2)$ ($x_1x_2 \neq 0$).

(I) If the line $l$ passes through the focus $F$ of the parabola, find the value of $x_1x_2$;

(II) If $x_1x_2 + y_1y_2 = 0$, find the value of $n$.

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