
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 equation of circle $C$ is $\begin{cases}x=2\cos \alpha+ \sqrt{3} \\ y=2\sin \alpha+1\end{cases} (\alpha$ is the parameter$)$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established.  
$(1)$ Write the polar equation of circle $C$ and the polar coordinates of the center $C$;  
$(2)$ The polar equation of line $l$ is $\theta= \frac{\pi}{3}(\rho\in R)$ and intersects circle $C$ at points $M$, $N$, find the area of $\triangle CMN$.

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