
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
The parametric equation of curve $C_{1}$ is $\begin{cases} x=2+2\cos \alpha \\ y=2\sin \alpha \end{cases}$ ($\alpha$ is the parameter), with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The curve $C_{2}$: $\rho=2\cos \theta$ intersects with the polar axis at points $O$ and $D$.  
(I) Write the polar equation of curve $C_{1}$ and the polar coordinates of point $D$;  
(II) The ray $l$: $\theta=\beta (\rho > 0, 0 < \beta < \pi)$ intersects with curves $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. Given that the area of $\triangle ABD$ is $\frac{\sqrt{3}}{2}$, find $\beta$.

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