
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]


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In the polar coordinate system, the equations of the lines on which the three sides of $\triangle OAB$ lie are $OA$: $\theta=0$, $OB$: $\theta= \frac{\pi}{2} $, $AB$: $\rho\cos (\theta- \frac{\pi}{3})= \sqrt{3} $, and $P$ is any point on the circumcircle $C$ of $\triangle OAB$. Establish a rectangular coordinate system with the polar point $O$ as the origin and the positive semi-axis of the polar axis as the $x$-axis using the same unit length.

(1) Find the coordinates of points $A$ and $B$ in the rectangular coordinate system and the parametric equation of circle $C$.

(2) Find the maximum and minimum values of $|PO|^2+|PA|^2+|PB|^2$.

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