
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 foci at $F_1$ and $F_2$, passing through the point $P(0, \sqrt{5})$, and having an eccentricity of $$\frac{2}{3}$$, let A be a moving point on the line $x = 4$.
(i) Find the equation of ellipse C.
(ii) If point B lies on ellipse C, such that $OA \perp OB$, find the minimum value of the length of segment AB.

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