
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 hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ $(a > 0,b > 0)$ with its right focus at $F$, and a circle passing through point $F$ with equation $(x-a)^2+y^2=\dfrac{c^2}{16}$ has a tangent line that is perpendicular to one of the asymptotes of $C$. Find the eccentricity of the hyperbola.

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