
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and passing through the point $P(0, \sqrt{5})$, with an eccentricity of $\frac{2}{3}$, and $A$ being a moving point on the line $x=4$.
- (I) Find the equation of the ellipse $C$;
- (II) Point $B$ is on the ellipse $C$, satisfying $OA \perpendicular OB$, find the minimum length of segment $AB$.

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