
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{4} = 1\ (a > 0)$, if a right-angled triangle is formed using its left and right foci as well as the point $(2, \sqrt{5})$ as its vertices, calculate the focal length of the hyperbola.

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