
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 curve $C$ as $\begin{cases} x=2\cos \alpha \\ y=1+2\sin \alpha \end{cases}$ (with $\alpha$ as the parameter), and the parametric equations of line $l$ as $\begin{cases} x=1+t\cos 45^\circ \\ y=t\sin 45^\circ \end{cases}$ (with $t$ as the parameter), establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis.  
(Ⅰ) Find the polar equation of curve $C$;  
(Ⅱ) Find the length of the chord cut from curve $C$ by line $l$.

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