
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0, b > 0$), with its two foci labeled $F_1$ and $F_2$, let point $P$ be on the left branch of the hyperbola. The line segment $PF_2$ intersects with the right branch of the hyperbola at point $Q$. If the triangle $PF_1Q$ is equilateral, calculate the eccentricity $e$ of this hyperbola.

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