
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
Consider the hyperbola $C:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\left( a > 0,b > 0 \right)$ with its left focus $F_{1}$. A line perpendicular to the $x$-axis passing through the right vertex intersects both asymptotes at points $A$ and $B$. If $\Delta AB{{F}\_{1}}$ is an equilateral triangle, find the eccentricity of $C$.

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