
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 $M(2, \sqrt{2})$ and $N(\sqrt{6}, 1)$, with $O$ as the coordinate origin:

1. Find the equation of ellipse $E$.
2. Determine if there exists a circle centered at the origin such that any tangent line to this circle intersects ellipse $E$ at two points $A$ and $B$, with $\overrightarrow{OA} \perp \overrightarrow{OB}$. If such a circle exists, write its equation; if not, explain why.

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