
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
Given the parametric equations of curve $C\_1$ as: $ \begin{cases} x=\cos \theta \\ y=1+\sin \theta \end{cases} (\theta \text{ is the parameter})$, establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of curve $C\_2$ is: $ρ=4\sin (θ+ \frac {π}{3})$, and the polar equation of line $l$ is $θ= \frac {π}{6}$.

1. Find the Cartesian equation of curve $C\_1$ and the polar equation of curve $C\_2$.
2. If curve $C\_1$ and curve $C\_2$ intersect with line $l$ at points $A$ and $B$ respectively (excluding the coordinate origin), find the value of $|AB|$.

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