
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 $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ where $a, b > 0$, and let O be the origin of the coordinate system.  
(1) Given that the ellipse E passes through points M(2, $\sqrt{2}$) and N($\sqrt{6}$, 1), find the equation of ellipse E;  
(2) If $a > b > 0$ and the foci are F<sub>1</sub>(-c, 0) and F<sub>2</sub>(c, 0), and M is a moving point on the ellipse such that $\overrightarrow {F_{1}M} \cdot \overrightarrow {F_{2}M} = 0$, find the range of the eccentricity of the ellipse;  
(3) Under the condition of (1), does there exist a circle with its center at the origin such that any tangent line to this circle intersects ellipse E at two points A and B, and $\overrightarrow {OA} \perp \overrightarrow {OB}$? If it exists, write down the equation of this circle and find the range of values for |AB|; if it does not exist, explain why.

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