
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 $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$ with a focal length of $2\sqrt{5}$, and one of its asymptotes is perpendicular to the line $2x+y=0$, determine the equation of the hyperbola.

A) $\frac{{x}^{2}}{4}-{y}^{2}=1$
B) ${x}^{2}-\frac{{y}^{2}}{4}=1$
C) $\frac{3{x}^{2}}{20}-\frac{3{y}^{2}}{5}=1$
D) $\frac{3{x}^{2}}{5}-\frac{3{y}^{2}}{20}=1$

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