
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, both of its foci and the endpoints of its minor axis lie on the circle $x^2 + y^2 = 1$.

(1) Find the equation of ellipse $C$.

(2) Suppose a line with slope $k$ passes through the point $M(2,0)$ and intersects the ellipse $C$ at points $A$ and $B$. Discuss for what values of $k$, $OA \perp OB$.

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