
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 rectangular coordinate system $(xOy)$, the parametric equations of curve $C\_1$ are given by: $\begin{cases} x=2+2\sqrt{5}\cos\alpha \\ y=4+2\sqrt{5}\sin\alpha \end{cases}$ ($\alpha$ is the parameter). Establish a polar coordinate system with coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of line $C\_2$ is $\theta=\frac{\pi}{3}$ ($\rho\in\mathbb{R}$).

(I) Find the polar equation of $C\_1$ and the rectangular equation of $C\_2$;

(II) If the polar equation of line $C\_3$ is $\theta=\frac{\pi}{6}$ ($\rho\in\mathbb{R}$), and the intersection points of $C\_2$ and $C\_1$ are $O$ and $M$, while the intersection points of $C\_3$ and $C\_1$ are $O$ and $N$, find the area of triangle $\triangle OMN$.

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