
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 Cartesian coordinate system $xOy$, the equation of line $l_{1}$ is $y= \sqrt {3}x$, and the parametric equation of curve $C$ is $\begin{cases} x=1+ \sqrt {3}\cos \varphi \\ y= \sqrt {3}\sin \varphi \end{cases}$ ($\varphi$ is the parameter, $0 \leqslant \varphi \leqslant \pi$). A polar coordinate system is established with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.  
$(1)$ Write the polar equations of line $l_{1}$ and curve $C$ respectively;  
$(2)$ If line $l_{2}: 2\rho\sin (\theta+ \frac {\pi}{3})+3 \sqrt {3}=0$, and the intersection point of line $l_{1}$ and curve $C$ is $A$, and the intersection point of line $l_{1}$ and $l_{2}$ is $B$, find $|AB|$.

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