
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
Consider the parabola $C$: $y^{2}=2px (p > 0)$ with focus $F$. Given that $P(2,m)$ is a point on the parabola $C$ and $|PF|=4$.
(I) Find the values of $p$ and $m$;
(II) Let the line $l_{1}$ passing through $Q(3,2)$ intersect the parabola $C$ at two points $A$ and $B$. The line $l_{2}$ passing through $F$ and perpendicular to $l_{1}$ intersects the parabola $C$ at points $M$ and $N$. If $|MN|$ is the geometric mean of $|QA|$ and $|QB|$, find $|MN|$.

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