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

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