
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 that point F is the right focus of the hyperbola $\frac{x^{2}}{a^{2}}\text{-}\frac{y^{2}}{b^{2}}\text{=}1(a\text{ > }0\text{,}b\text{ > }0)$, point A is a vertex on the imaginary axis of the hyperbola, and line FA intersects with one asymptote of the hyperbola at point B on the left side of the y-axis. If $\overrightarrow{\text{FA}}\text{=}(\sqrt{3}\text{-}1)\overrightarrow{\text{AB}}$, find the eccentricity of this hyperbola.

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