
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
In the Cartesian coordinate system $xoy$, a line $m$ passes through a fixed point $C(p, 0)$ and intersects the parabola $y^2 = 2px$ ($p > 0$) at points $A$ and $B$.
(I) Let $N(-p, 0)$, find the minimum value of $\overrightarrow{NA} \cdot \overrightarrow{NB}$;
(II) Is there a line $l$ perpendicular to the $x$-axis, such that the length of the chord intercepted by the circle with diameter $AC$ is always a constant? If it exists, find the equation of $l$; if not, explain why.

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