
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 $E$ with eccentricity $\frac{\sqrt{2}}{2}$ and the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) whose focal distance is 4.  
(1) Find the equation of ellipse $E$;  
(2) Suppose there is a circle with its center at the origin $O$, and any tangent line to this circle intersects ellipse $E$ at two points $A$ and $B$, with $\overrightarrow{OA} \perp \overrightarrow{OB}$. Find the equation of this circle and the maximum value of $|AB|$.

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