
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 $C$: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ with its left and right foci being $F_1$ and $F_2$ respectively, and an equilateral triangle $AF_1F_2$ with one side $AF_1$ intersecting the left branch of the hyperbola at point $B$, and $\vec{AF_1} = 4\vec{BF_1}$, find the eccentricity of the hyperbola $C$.

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