
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
【Elective 4-4: Coordinate System and Parametric Equations】Given the parametric equations of line $l$ as $\begin{cases}x=4+ \frac{1}{2}t \\ y= \frac{ \sqrt{3}}{2}t\end{cases} (t$ is the parameter$)$, with the coordinate origin as the pole and the positive half-axis of $x$ as the polar axis to establish a polar coordinate system, the polar equation of curve $C$ is $\rho=2\cos \theta$.

(Ⅰ) Find the Cartesian coordinate equation of curve $C$ and the polar coordinate equation of line $l$;

(Ⅱ) If the line $\theta= \frac{\pi}{6} (\rho\in\mathbb{R} )$ intersects curve $C$ at point $A$ (different from the origin), and intersects line $l$ at point $B$, find $|AB|$.

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