
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 equation of line $l$ as $\begin{cases} x=t\cos \alpha \\ y=1+t\sin \alpha \end{cases}$ (where $t$ is the parameter, $\frac{\pi}{2} \leqslant \alpha < \pi$), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of circle $C$ is $\rho=2\cos \theta$.  
(Ⅰ) Discuss the number of common points between line $l$ and circle $C$;  
(Ⅱ) Draw a perpendicular line from the pole to line $l$, with the foot of the perpendicular being $P$, and find the length of the chord obtained by the intersection of the trajectory of point $P$ with circle $C$.

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