
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 with eccentricity $2$ and equation $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, the right focus $F_2$ of the hyperbola is the focus of the parabola $y^2 = 8x$. A line $l$ passing through point $F_2$ intersects the right branch of the hyperbola at two points $P$ and $Q$. $F_1$ is the left focus of the hyperbola. If $PF_1 \perp QF_1$, then find the slope of line $l$.

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