
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 parabola $y^{2}=8x$, a circle with the focus of the parabola as its center, and a hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a>0, b>0)$, where the radius of the circle is the length of the imaginary semi-axis $b$ of the hyperbola, and the circle is tangent to one of the asymptotes of the hyperbola. When $\frac{4}{a^{2}}+\frac{1}{b^{2}}$ reaches its minimum value, find the eccentricity of the hyperbola.

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