
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with its foci on the $x$-axis, and the left vertex of the ellipse $E$ is $A$. A line with slope $k(k > 0)$ intersects the ellipse $E$ at points $A$ and $B$. Point $C$ is on the ellipse $E$, and $AB \perpendicular AC$. The line $AC$ intersects the $y$-axis at point $D$.

(Ⅰ) When point $B$ is the top vertex of the ellipse, and the area of $\triangle ABD$ is $2ab$, find the eccentricity of the ellipse;

(Ⅱ) When $b= \sqrt{3}$ and $2|AB|=|AC|$, find the range of $k$.

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