
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
The equation $\frac{x^2}{a} + \frac{y^2}{b} = 1$ represents an ellipse. The eccentricity of an ellipse is given by $e = \sqrt{1-\frac{b^2}{a^2}}$, where $b$ and $a$ are the lengths of the semi-minor and semi-major axes, respectively. Given that the arithmetic mean of two positive numbers $1$ and $9$ is $a$ and the geometric mean is $b$, determine the eccentricity of the curve.

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