
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: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{3}}{2}$, and the area of the rhombus formed by the four vertices of the ellipse is 4.  
(Ⅰ) Find the equation of the ellipse;  
(Ⅱ) The vertices of quadrilateral ABCD are on ellipse C, and the diagonals AC, BD both pass through the origin O. If $k_{AC} \cdot k_{BD} = -\frac{1}{4}$,  
(i) Find the range of $\overrightarrow{OA} \cdot \overrightarrow{OB}$; (ii) Find the area of quadrilateral ABCD.

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