
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 given by the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ ($a > b > 0$), and a circle O with the equation $x^2 + y^2 = r^2$ ($0 < r < b$). If a tangent line l of circle O, described by the equation $y = kx + m$, intersects the ellipse E at points A and B:
(Ⅰ) When $k = -\frac{1}{2}$ and $r = 1$, if points A and B both lie on the positive half of the coordinate axes, find the equation of the ellipse E;
(Ⅱ) If the circle with diameter AB passes through the origin O, investigate the relationship between $a$, $b$, and $r$, and explain the reasoning.

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