
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 rectangular coordinate system $(xOy)$, the parametric equations of the curve $C\_1$ are given by $\begin{cases} x=2+2\cos \phi \\ y=2\sin \phi \end{cases}$ ($\phi$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the curve $C\_2$ is given by $\rho=4\sin \theta$.
(I) Find the Cartesian equation of $C\_1$ and the polar equation of $C\_2$.
(II) Given that the polar equation of the curve $C\_3$ is $\theta=\alpha$ ($0 < \alpha < \pi$, $\rho\in \mathbb{R}$), point $A$ is the intersection of curves $C\_3$ and $C\_1$, point $B$ is the intersection of curves $C\_3$ and $C\_2$, and both points $A$ and $B$ are distinct from the origin $O$ with $|AB|=4\sqrt{2}$, find the value of $\alpha$.

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