
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 $\begin{cases} x=3-\frac{\sqrt{2}}{2}t \
y=\sqrt{5}+\frac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter). In the polar coordinate system, which has the same length unit as the rectangular coordinate system $(xOy)$, with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the equation of circle $C$ is $\rho =2\sqrt{5}\sin \theta$.

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

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