
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 a hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with left and right foci $F_1$ and $F_2$, respectively, a circle centered at $F_2$ intersects the hyperbola $C$ at point $P$ in the first quadrant. The line $PF_1$ is tangent to the circle at point $P$ and intersects the left branch of the hyperbola at point $Q$. Additionally, $|PQ| = 2|F_1Q|$. Determine the eccentricity of the hyperbola.

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