
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 circle $C$ are: $$\begin{cases} x=2+2\cos\theta \\ y=2\sin\theta \end{cases}$$ (where $\theta$ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the non-negative semi-axis of $x$ as the polar axis. The polar equation of ray $l$ is $\theta=\alpha$ ($\rho>0$).
(1) Convert the parametric equations of circle $C$ to polar coordinates;
(2) Suppose point $A$ has rectangular coordinates $(1, \sqrt{3})$, and ray $l$ intersects circle $C$ at point $B$ (which is not point $O$). Find the maximum area of $\triangle OAB$.

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