
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through $(0, \sqrt{2})$, and the eccentricity of ellipse $C$ is $\frac{\sqrt{2}}{2}$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ Suppose the line $l$, which has a defined slope, intersects ellipse $C$ at points $P$ and $Q$, $O$ is the origin, $OP \perp OQ$, and $l$ is tangent to a fixed circle $W$ with center $O$, find the equation of circle $W$.

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