
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 parabola $C$: $y^{2}=2px$ with the focus at $F(2,0)$, and points $P(m,0)$ and $Q(-m,n)$, a line $l$ passing through $P$ with a slope of $k$ (where $k\neq 0$) intersects the parabola $C$ at points $A$ and $B$.
(Ⅰ) For $m=k=2$, if $\vec{QA} \cdot \vec{QB} = 0$, find the value of $n$.
(Ⅱ) If $O$ represents the origin and $m$ is constant, for any change in $k$ such that $\vec{OA} \cdot \vec{OB} = 0$ always holds, find the value of the constant $m$.
(Ⅲ) For $k=1$, $n=0$, and $m < 0$, find the maximum area of triangle $QAB$ as $m$ changes.

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