
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 $\begin{cases} x=\sqrt{7}\cos \alpha \\ y=2+\sqrt{7}\sin \alpha \\ \end{cases}$ (where $\alpha$ is a parameter), and the curve $C\_2: (x-1)^2+y^2=1$. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive half of the $x$-axis as the polar axis.

(1) Find the Cartesian equation of curve $C\_1$ and the polar equation of curve $C\_2$;

(2) If the ray $\theta = \frac{\pi }{6}(\rho > 0)$ intersects 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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