
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
Two positive numbers, \(a\) and \(b\), have \(\frac{5}{2}\) as their arithmetic mean and \(\sqrt{6}\) as their geometric mean, with \(a > b\). Determine the eccentricity \(e\) of the hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

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