
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 $xOy$, there is a circle $C_{1}$: $(x-2)^{2}+(y-4)^{2}=20$. With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. For $C_{2}$: $\theta= \frac {\pi}{3}(\rho\in\mathbb{R})$.

$(1)$ Find the polar equation of $C_{1}$ and the Cartesian coordinate equation of $C_{2}$;  
$(2)$ If the polar equation of line $C_{3}$ is $\theta= \frac {\pi}{6}(\rho\in\mathbb{R})$, and assuming the intersection points of $C_{2}$ and $C_{1}$ are $O$ and $M$, and the intersection points of $C_{3}$ and $C_{1}$ are $O$ and $N$, find the area of $\triangle OMN$.

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