
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
Consider the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) with vertices $A(a,0)$, $B(0,b)$, and left focus $F$. If $\triangle FAB$ is a right triangle with the right angle at $B$, find the eccentricity $e$ of the ellipse.

A) $\frac{\sqrt{3}-1}{2}$
B) $\frac{1+\sqrt{5}}{4}$
C) $\frac{\sqrt{5}-1}{2}$
D) $\frac{1+\sqrt{3}}{4}$

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