
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 curve $C$ is given by the parametric equations $\begin{cases} x= \sqrt {3}+2\cos \alpha \\ y=1+2\sin \alpha\end{cases}$ (where $\alpha$ is the parameter). A polar coordinate system is established with the origin of the Cartesian coordinate system as the pole and the positive $x$-axis as the polar axis.  
$(1)$ Find the polar equation of curve $C$;  
$(2)$ Lines $l_{1}$ and $l_{2}$ pass through the origin $O$ and intersect curve $C$ at points $A$ and $B$ other than the origin. If $\angle AOB= \dfrac {\pi}{3}$, find the maximum value of the area of $\triangle AOB$.

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