
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 an ellipse E: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) that passes through points $M(2, \sqrt {2})$ and $N(\sqrt {6}, 1)$, with $O$ as the origin of the coordinate system,
(1) Find the equation of the ellipse E;
(2) Determine whether there exists a circle with the center at the origin such that any tangent line to the circle intersects the ellipse E at two points $A$ and $B$, and also $\overrightarrow {OA}\perp \overrightarrow {OB}$. If such a circle exists, write its equation and find the range of the length $|AB|$; if not, explain why.

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