
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 parametric equation of curve $C_{1}$ is $\begin{cases} x=1+\cos \theta \\ y=\sin \theta \end{cases}$ (where $\theta$ is the parameter), with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C_{2}$ is $\rho=\sin \theta+\cos \theta$, and the polar equation of curve $C_{3}$ is $\theta= \frac {\pi}{6}$.

$(1)$ Convert the parametric equation of curve $C_{1}$ into a polar equation;  
$(2)$ Curve $C_{3}$ intersects curve $C_{1}$ at points $O$ and $A$, and curve $C_{3}$ intersects curve $C_{2}$ at points $O$ and $B$. Find $|AB|$.

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