
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 line $l:3x-\sqrt{3}y-6=0$, and the curve $C:\rho -4\sin \theta =0$ in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.

(I) Write the line $l$ in parametric form $\begin{cases} x=2+t\cos \alpha \\ y=t\sin \alpha \end{cases}(t$ is the parameter, $\alpha \in \left[ 0,\pi \right))$ and find the rectangular coordinate equation of the curve $C$;

(II) Draw a line with a slope of $30^{\circ}$ through any point $P$ on the curve $C$, intersecting $l$ at point $A$, and find the maximum value of $|AP|$.

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