
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 curve $C_{1}$ is given by the parametric equations $\begin{cases} x=1+\cos \theta \\ y=\sin \theta \end{cases}$ (with $\theta$ as the parameter). Taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of the curve $C_{2}$ is $\rho^{2}-4\rho\sin \theta=3$.  
$(1)$ Find the polar equation of the curve $C_{1}$ and the Cartesian equation of the curve $C_{2}$.  
$(2)$ The line $\theta= \frac {\pi}{3}$ intersects the curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ in the first quadrant, respectively. Calculate $|AB|$.

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