
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
Given the parametric equations of curve $C$ in the Cartesian coordinate system $xOy$ as $\begin{cases} x = 1 + \sqrt{3} \cos \theta \\ y = \sqrt{3} \sin \theta \end{cases}$, where $\theta$ is the parameter. Establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.

(I) Find the ordinary equation and the polar equation of curve $C$.
(II) The polar equation of line $l$ is $\rho \cos (\theta - \frac{\pi}{6}) = 3\sqrt{3}$. Ray $OT: \theta = \frac{\pi}{3} (\rho > 0)$ intersects curve $C$ at point $A$ and line $l$ at point $B$. Find the length of segment $AB$.

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