
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 that $F$ is the right focus of the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a > 0, b > 0\right)$, $A$ is the right vertex of $C$, $B$ is a point on $C$, and $BF$ is perpendicular to the $x$-axis. If the slope of $AB$ is $3$, then the eccentricity of $C$ is ____.

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