
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 $$\begin{cases} \overset{x=1+cos\alpha }{y=sin\alpha }\end{cases}$$ (where $\alpha$ is the parameter). Establish a polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of the curve $C\_2$ is $\rho=2 \sqrt {3}sin\theta$.
(1) Find the polar equation of the curve $C\_1$ and the rectangular equation of the curve $C\_2$;
(2) Suppose the moving line $l$: $y=kx$ ($x\neq0$, $k\neq0$) intersects with curves $C\_1$ and $C\_2$ at points $A$ and $B$, respectively. Find the value of $k$ when $|AB|$ reaches the maximum, and find the maximum value of $|AB|$.

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