
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 $C: \dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\left(a \gt 0,b \gt 0\right)$, both of its asymptotes are tangent to the circle $\left(x-a\right)^{2}+y^{2}=\dfrac{b^{2}}{4}$. Determine the eccentricity of the hyperbola $C$.

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