
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 the line $l$ is
$$
\begin{cases}
x=3-\frac{\sqrt{2}}{2}t \\
y=\sqrt{5}-\frac{\sqrt{2}}{2}t
\end{cases}
$$
(where $t$ is the parameter). In the polar coordinate system (using the same unit length as the Cartesian coordinate system $xOy$, with the origin $O$ as the pole, and the positive $x$-axis as the polar axis), the equation of the circle $C$ is $\rho=2\sqrt{5}\sin\theta$.
(Ⅰ) Find the Cartesian equation of circle $C$;
(Ⅱ) Let circle $C$ intersect with line $l$ at points $A$ and $B$. Given that 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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