
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 $x^{2} - \frac{y^{2}}{b^{2}} = 1$ with $b > 0$, and a circle centered at origin $O$ with a radius equal to the real semi-axis of the hyperbola. The circle intersects the two asymptotes of the hyperbola at points $A$, $B$, $C$, $D$. The area of the quadrilateral formed by these four points is $b$. Find the eccentricity of the hyperbola.

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

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