
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 left focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$ and $b > 0$), point $A$ is a vertex on the imaginary axis of the hyperbola, and the line passing through points $F$ and $A$ intersects one asymptote of the hyperbola on the right side of the $y$-axis at point $B$. If $\vec{FA} = (\sqrt{2} - 1)\vec{AB}$, then the eccentricity of this hyperbola is ______.

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