
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
In the Cartesian coordinate system $(xOy)$, the parametric equations of curve $C\_1$ are given by $\begin{cases} x=2\cos \alpha +1 \\\\ y=2\sin \alpha \end{cases}$ where $\alpha$ is the parameter. In the polar coordinate system established with the origin $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis, the polar equation of curve $C\_2$ is given by $\rho=4\sin \theta$.

(1) Find the Cartesian equation of curve $C\_1$ and the polar equation of curve $C\_2$;
(2) Find the length of the common chord of curves $C\_1$ and $C\_2$.

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