
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 the ellipse $E: \frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0 )$ passing through points $M\left(2, \sqrt{2}\right),N\left( \sqrt{6},1\right)$, with $O$ representing the coordinate origin.

(I) Find the equation of ellipse $E$;
(II) Determine whether there exists a circle centered at the origin such that any tangent line to this circle intersects with ellipse $E$ at two points $A,B$, and $OA \perp OB$. If such a circle exists, find its equation; if not, explain the reason.

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