
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 left and right foci $F_{1}$ and $F_{2}$, respectively. The ellipse passes through point $P(0, \sqrt{5})$, has an eccentricity of $\frac{2}{3}$, and the line $l$ passing through $F_{1}$ intersects the line $x=4$ at point $A$.

(I) Find the equation of the ellipse $C$;
(II) When the perpendicular bisector of line segment $F_{1}A$ passes through point $F_{2}$, find the equation of line $l$;
(III) Point $B$ is on the ellipse $C$, and $OA \perp OB$, find the minimum length of line segment $AB$.

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