
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 $\frac{{x}^{2}}{{a}^{2}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$, one of its asymptotes is parallel to the line $l: x + 3y + 2\sqrt{5} = 0$, and one of the hyperbola's foci lies on line $l$. Determine the equation of the hyperbola.

A) $\frac{{{x}^{2}}}{18}-\frac{{{y}^{2}}}{2}=1$
B) $\frac{{{x}^{2}}}{2}-\frac{{{y}^{2}}}{18}=1$
C) $\frac{{{x}^{2}}}{20}-\frac{{{y}^{2}}}{5}=1$
D) $\frac{{{x}^{2}}}{25}-\frac{{{y}^{2}}}{100}=1$

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