
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 hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), perpendiculars are drawn from the right focus $F(2\sqrt{2}, 0)$ to the two asymptotes, with the feet of the perpendiculars being $A$ and $B$, respectively. Let point $O$ be the origin. If the area of quadrilateral $OAFB$ is $4$, determine the eccentricity of the hyperbola.

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