
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 curve $C$ is given by the parametric equations 
$$
\begin{cases} 
x = 2 + 2\sqrt{2}\cos \theta, \\ 
y = -2 + 2\sqrt{2}\sin \theta 
\end{cases}
\text{where } \theta \text{ is the parameter.}
$$
The origin $O$ is taken as the pole and the positive $x$-axis as the initial line for a polar coordinate system.

(I) Find the polar equation of the curve $C$.

(II) A line $l$ with a slope of $1$ passes through point $P(2,0)$ and intersects the curve $C$ at points $A$ and $B$. Find the value of $|PA|+|PB|$.

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