
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 Cartesian coordinate system $xOy$, with $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of circle $C$ is $\rho=4\sin\theta$, and the parametric equation of line $l$ is $\begin{cases}x=- \frac{ \sqrt{3}}{2}t \\ y=2+ \frac{t}{2}\end{cases}$ (where $t$ is the parameter). Line $l$ intersects circle $C$ at points $A$ and $B$.  
$(1)$ Find the polar coordinates of the center of the circle;  
$(2)$ The intersection of line $l$ with the $x$-axis is point $P$, find $|PA|+|PB|$.

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