
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: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ (where $a>b>0$) that passes through the point $E(\sqrt 3, 1)$, with an eccentricity of $\frac{\sqrt{6}}{3}$, and O as the origin of the coordinate system.
(Ⅰ) Find the equation of the ellipse C.
(Ⅱ) If point P is a moving point on ellipse C, and the perpendicular bisector of AP, with A(3, 0), intersects the y-axis at point B, find the minimum value of |OB|.

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