
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 $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, a circle with center $F_2$ and radius $|OF_2|$ is drawn. A tangent line is drawn from point $F_1$ to the circle, intersecting the circle at point $Q$. If the line segment $F_1Q$ is bisected by one of the hyperbola's asymptotes, determine the eccentricity of the hyperbola.

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