
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 $xoy$, the parametric equation of line $l$ is $\begin{cases}x=3- \dfrac{ \sqrt{2}}{2}t \\ y= \sqrt{5}+ \dfrac{ \sqrt{2}t}{2}\end{cases}$ ($t$ is the parameter), in the polar coordinate system (with the same unit length as the Cartesian coordinate system $xoy$, and the origin $O$ as the pole, and the positive half-axis of $x$ as the polar axis), the equation of circle $C$ is $\rho=2 \sqrt{5}\sin \theta$.

$(1)$ Find the distance from the center of circle $C$ to line $l$;

$(2)$ Suppose circle $C$ intersects line $l$ at points $A$ and $B$. If the coordinates of point $P$ are $(3, \sqrt{5})$, find $|PA|+|PB|$.

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