
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 the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ passing through the point $E(\sqrt{3}, 1)$, with an eccentricity of $\frac{\sqrt{6}}{3}$, and $O$ as the coordinate origin.
(I) Find the equation of the ellipse $C$;
(II) If point $P$ is a moving point on the ellipse $C$, and the perpendicular bisector of segment $AP$, where $A(3, 0)$, 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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