
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
In the rectangular coordinate system $(xOy)$, the parametric equation of line $l$ is given by $ \begin{cases} x=-\frac{1}{2}t \\ y=2+\frac{\sqrt{3}}{2}t \end{cases} (t\text{ is the parameter})$, and a circle $C$ with polar coordinate equation $\rho=4\cos\theta$ is established with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. Let $M$ be any point on circle $C$, and connect $OM$ and extend it to $Q$ such that $|OM|=|MQ|$.
(I) Find the rectangular coordinate equation of the trajectory of point $Q$;
(II) If line $l$ intersects the trajectory of point $Q$ at points $A$ and $B$, and the rectangular coordinates of point $P$ are $(0,2)$, find the value of $|PA|+|PB|$.

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