
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
Selected Exercise $(4-4)$: Coordinate Systems and Parametric Equations

Given that the parametric equation of line $l$ is $\begin{cases} & x=3+t \\ & y=\sqrt{5}+t \\ \end{cases}$ ($t$ is the parameter), and the polar coordinate equation of circle $C$ is $\rho =2\sqrt{5}\sin \theta$.

$(1)$ Find the rectangular coordinate equation of circle $C$;

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

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