
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 that the parametric equation of line $(1)$ is $ \begin{cases} x=3-t \\ y= \sqrt {5}+t \end{cases} (t $ is the parameter$)$, a polar coordinate system is established with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of circle $C$ is $ρ=2 \sqrt {5}\sin θ$.
$(1)$ Find the rectangular coordinate equation of circle $C$;
$(2)$ Suppose circle $C$ intersects with line $l$ at points $A$ and $B$, and point $P$ has coordinates $(3, \sqrt {5})$, find $|PA|+|PB|$.

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