
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, the curve $C_{1}$ is defined by $x^{2}-y^{2}=2$, and the parametric equations of curve $C_{2}$ are $\left\{\begin{array}{l}x=2+2\cos\theta\\ y=2\sin\theta\end{array}\right.$ ($\theta$ is the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equations of curves $C_{1}$ and $C_{2}$;<br/>$(2)$ In the polar coordinate system, the ray $\theta=\frac{\pi}{6}$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ (other than the pole $O$), find the length of $|AB|$.

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