
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
A line perpendicular to the line $y=-\frac{b}{a}x$ passing through the right focus $F$ of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a > 0, b > 0$) intersects the hyperbola at point $B$ on its left branch, and the foot of the perpendicular is $A$. If $\vec{FB}=2\vec{FA}$, then find the eccentricity of the hyperbola.

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