
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 $\frac{x^2}{4} - \frac{y^2}{b^2} = 1 (b > 0)$, a circle with the origin as its center and the real semi-axis length of the hyperbola as its radius intersects with the two asymptotes of the hyperbola at points $A$, $B$, $C$, and $D$. The area of the quadrilateral $ABCD$ is $2b$. Determine the equation of the hyperbola.

A) $\frac{x^2}{4} - \frac{3y^2}{4} = 1$
B) $\frac{x^2}{4} - \frac{4y^2}{3} = 1$
C) $\frac{x^2}{4} - \frac{y^2}{10} = 1$
D) $\frac{x^2}{4} - \frac{y^2}{12} = 1$

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