
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 a hyperbola $F$ with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $F$ be its left focus. Draw a line perpendicular to one asymptote of the hyperbola passing through $F$, and denote the foot of the perpendicular as $A$ and the intersection with the other asymptote as $B$. If $3\overrightarrow{FA} = \overrightarrow{FB}$, find the eccentricity of this hyperbola.

A) $2$
B) $3$
C) $\sqrt{2}$
D) $\sqrt{3}$

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