
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 parametric equations of line $l$ as $\begin{cases}x=t \\ y=mt\end{cases} (t$ is the parameter$)$, and the parametric equations of circle $C$ as $\begin{cases}x=\cos \alpha \\ y=1+\sin \alpha\end{cases} (\alpha$ is the parameter$)$.

(Ⅰ) If the chord length of the intersection between line $l$ and circle $C$ is not less than $\sqrt{2}$, find the range of the real number $m$;

(Ⅱ) If the coordinates of point $A$ are $(2,0)$, and the moving point $P$ is on circle $C$, try to find the equation of the trajectory of the midpoint $Q$ of segment $PA$.

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