
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 with the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a, b > 0)$, let $F$ be its right focus, and draw a line perpendicular to one of its asymptotes passing through $F$. Let $P$ be the foot of the perpendicular, and let $Q$ be the intersection of the perpendicular bisector of $OP$ with the $y$-axis, where $O$ is the origin. If the area of triangle $OFQ$ is four times the area of triangle $OPQ$, find the eccentricity of the hyperbola.

A) $\sqrt{2}$
B) $\sqrt{3}$
C) $2$
D) $\sqrt{5}$

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