
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 the parabola $C: y^2 = 2px$ ($p > 0$) with focus $F$ and the distance from $F$ to the asymptotes of the hyperbola $x^2 - \frac{y^2}{3} = 1$ is $\sqrt{3}$, a line passing through the focus $F$ with slope $k$ intersects the parabola $C$ at points $A$ and $B$, and $\overrightarrow{AF} = 2\overrightarrow{FB}$, determine $|k|$.

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