
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
Let $F$ be the right focus of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$). A line passing through the origin intersects the left and right branches of the hyperbola at points $P$ and $Q$, respectively. If $|PQ| = 2|QF|$ and $\angle PQF = 60^{\circ}$, calculate the eccentricity of this hyperbola.

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