
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 equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=3\cos\varphi}\\{y=3\sin\varphi+3}\end{array}\right.$ ($\varphi$ is the parameter). Taking the coordinate origin $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, establish a polar coordinate system. The polar coordinate equation of curve $C_{2}$ is $\rho \sin (\theta-\frac{\pi}{3})=2$.  
$(1)$ Write down the polar coordinate equation of $C_{1}$ and the Cartesian equation of $C_{2}$.  
$(2)$ Let curve $C_{3}$: $\theta =\frac{5\pi}{6}$ ($\rho\ \gt 0$) intersect $C_{1}$ and $C_{2}$ at points $M$ and $N$ respectively. Find the value of $|MN|$.

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