
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 curve $C\_1$ are given by $\begin{cases}x=2\cos \alpha \ y=2+2\sin \alpha\end{cases} (\alpha$ is the parameter). Point $M$ is a moving point on $C\_1$, and point $P$ satisfies $\overrightarrow{OP} =2 \overrightarrow{OM}$. The trajectory of point $P$ is curve $C\_2$.

(I) Find the equation of $C\_2$;

(II) In the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the ray $\theta= \frac{\pi}{3} $ intersects $C\_1$ at point $A$ and intersects $C\_2$ at point $B$. Find $|AB|$.

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