
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 curve $C_1$ is 
$$
\begin{cases} 
x=2\sqrt{2}-\frac{\sqrt{2}}{2}t \\
y=\sqrt{2}+\frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\sqrt{2}\sin \theta$.

(Ⅰ) Convert the equation of $C_2$ into a Cartesian coordinate equation;

(Ⅱ) Suppose $C_1$ and $C_2$ intersect at points $A$ and $B$, and the coordinates of point $P$ are $(\sqrt{2},2\sqrt{2})$, find $|PA|+|PB|$.

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