
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 a line $l: 3x - \sqrt{3}y - 6 = 0$ and a polar coordinate system with the origin $O$ as the pole and the positive $x$-axis as the polar axis, consider the curve $C: \rho - 4\sin \theta = 0$.
(Ⅰ) Convert the line $l$ into a parametric equation form $\begin{cases} x=2+t\cos \alpha \\ y=t\sin \alpha \end{cases}$ (where $t$ is the parameter, $\alpha \in [0,\pi)$), and find the Cartesian coordinate equation of curve $C$.
(Ⅱ) For any point $P$ on curve $C$, construct a line with a slope angle of $30^{\circ}$ that intersects line $l$ at point $A$. Determine the maximum and minimum values of $|AP|$.

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