
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 equation of line $l$ as $\begin{cases}x= \frac{1}{2}t \\ y= \frac{ \sqrt{3}}{2}t\end{cases} (t\text{ is the parameter})$ and the parametric equation of curve $C$ as $\begin{cases}x=1+2\cos θ \\ y=2 \sqrt{3}+2\sin θ\end{cases} (θ\text{ is the parameter})$, and point $P$ has polar coordinates $(2 \sqrt{3} , \frac{2π}{3} )$ in a polar coordinate system established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis.

(I) Find the polar coordinate equations of line $l$ and curve $C$.
(II) Suppose line $l$ intersects curve $C$ at points $A$ and $B$. Find the area of $\triangle PAB$.

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