
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 plane rectangular coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=1+\cos\alpha,\\ y=\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinate equation of curve $C_{2}$ is $\rho =-2\sin \theta$.  
$(1)$ Find the polar coordinate equation of curve $C_{1}$ and the rectangular coordinate equation of curve $C_{2}$;  
$(2)$ Suppose the line $l: \sqrt{3}x+y=0$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ (other than the pole), find the length of line segment $AB$.

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