
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 $E$ have its center at the origin, with foci on the coordinate axes. Ellipse $E$ passes through points $M(2,\sqrt{2})$ and $N(\sqrt{6},1)$. Let $O$ be the origin.

$(1)$ Find the equation of ellipse $E$;

$(2)$ Is there 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; if not, explain why.

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