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

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