
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 eccentricity $$\frac{\sqrt{3}}{2}$$, and the distance from its left vertex to the line $x + 2y - 2 = 0$ is $$\frac{4\sqrt{5}}{5}$$.

(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) Suppose line $l$ intersects ellipse C at points A and B. If the circle with diameter AB passes through the origin O, investigate whether the distance from point O to line AB is a constant. If so, find this constant; otherwise, explain why;  
(Ⅲ) Under the condition of (Ⅱ), try to find the minimum value of the area $S$ of triangle $\triangle AOB$.

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