
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
Given a coordinate system and parametric equations of a circle $C: \begin{cases} x=2+\sqrt{2}\cos \theta, \\ y=2+\sqrt{2}\sin \theta \end{cases} (\theta$ is the parameter), and establishing a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the polar coordinates of points $A$ and $B$ are respectively $(1,\pi)$ and $(1,0)$.

$(1)$ Find the polar equation of circle $C$;

$(2)$ If $P$ is a moving point on circle $C$, find the range of values for $|PA|^2+|PB|^2$.

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