
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 curve $C$ in the Cartesian coordinate system $Oxy$ has the parametric equations $\begin{cases} x=1+\sqrt{3}\cos {\theta} \\ y=\sqrt{3}\sin {\theta} \end{cases}$ (where $\theta$ is the parameter), establish a polar coordinate system with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis.

(Ⅰ) Find the polar equation for curve $C$;

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

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