
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]


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Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with one asymptote line $y = kx (k > 0)$, an eccentricity $e = \sqrt{5}k$, and passing through point $P(4, 1)$, find the equation of the hyperbola. The options are:
A) $\frac{x^2}{3} - \frac{y^2}{12} = 1$
B) $x^2 - \frac{y^2}{4} = 1$
C) $\frac{x^2}{12} - \frac{y^2}{3} = 1$
D) $\frac{x^2}{4} - y^2 = 1$

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