
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
Let ellipse $E$ be defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a,b>0$) and let points $M(2, \sqrt{2})$, $N(\sqrt{6}, 1)$, with $O$ being the origin.  
(Ⅰ) Find the equation of ellipse $E$;  
(Ⅱ) Determine if there exists a circle centered at the origin such that any tangent line to this circle intersects ellipse $E$ at exactly two points $A$ and $B$, and $\overrightarrow{OA} \perp \overrightarrow{OB}$. If such a circle exists, write down its equation and the range of values for $|AB|$; if not, explain why.

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