
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 $\begin{cases}x=1+ \sqrt{3}\cos \theta \
y= \sqrt{3}\sin \theta\end{cases}$, where $\theta$ is a parameter. Point $M$ is a moving point on curve $C\_1$, and point $P$ is on curve $C\_2$ such that $\overrightarrow{OP}=2 \overrightarrow{OM}$.
(I) Find the Cartesian equation of curve $C\_2$.
(II) Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. Let the ray $\theta= \frac{\pi}{3}$ intersect curves $C\_1$ and $C\_2$ at points $A$ and $B$, respectively. Find $|AB|$.

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