
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 plane $(xOy)$, a circle $⊙C$ passes through the three intersection points of the quadratic function $f(x)=\frac{\sqrt{3}}{3}(x^2+2x-3)$ and the two coordinate axes.
(1) Find the standard equation of $⊙C$;
(2) Given point $A(-2,0)$ and point $B(2,0)$, investigate whether there exists a point $P$ on $⊙C$ such that $PA=\sqrt{2}PB$. If it exists, find the coordinates of point $P$; if not, explain the reason.

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