
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 a known ellipse $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, and a circle $O$: $x^2 + y^2 = r^2$ $(0 < r < b)$. A tangent line $l$ to circle $O$, $y = kx + m$, intersects with ellipse $E$ at points $A$ and $B$.
(Ⅰ) When $k = -\frac{1}{2}$ and $r = 1$, if points $A$ and $B$ are both on the positive half-axes, find the equation of ellipse $E$;
(Ⅱ) If the circle with $AB$ as its diameter goes through the origin $O$, explore the relationship among $a$, $b$, and $r$, and explain your reasoning.

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