
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
In the Cartesian coordinate system $xOy$, there is an ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and a circle $O$: $x^2 + y^2 = r^2 (0 < r < b)$. When a tangent line $l$ of circle $O$: $y = kx + m$ intersects with ellipse $E$ at points $A$ and $B$.

(Ⅰ) When $k = -\frac{1}{2}$, $r = 1$, if points $A$ and $B$ are both on the positive axes, find the equation of ellipse $E$;

(Ⅱ) If the circle with diameter $AB$ passes through the origin $O$, investigate whether $a$, $b$, $r$ satisfy $\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{r^2}$, and explain the reason.

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