
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 ${C}_{1}:{x}^{2}-{y}^{2}=1$, the parameter equation of curve $C_{2}$ is $\left\{\begin{array}{l}x=1+cosα,\\ y=sinα\end{array}\right.$ ($\alpha$ is the parameter), with the coordinate origin $O$ as the pole, and the positive half-axis of the $x$-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 ${θ}_{0}=\frac{π}{6}(ρ≥0)$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ (different from the pole $O$), find $|AB|$.

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