
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 $C$ are $\left\{\begin{array}{l}x=2+2\cos\alpha,\\ y=2\sin\alpha\end{array}\right.$ ($\alpha$ is the parameter), with $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis.  
$(1)$ Find the polar coordinate equation of curve $C$;  
$(2)$ The polar coordinate equation of curve $C_{1}$ is $ρ(\sqrt{3}\cos\theta-\sin\theta)=4$, and the polar coordinate equation of curve $C_{2}$ is $θ=\frac{π}{6}$. One intersection point of $C_{2}$ and $C$ is $M$ (point $M$ is different from point $O$), and the intersection point with $C_{1}$ is $N$. Find $|MN|$.

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