
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
A particle is placed on the curve $y = x^3 - x^2 - x - 6$ at a point $P$ whose $y$-coordinate is $8$. Find the horizontal distance traveled by the particle as it rolls along the curve until it reaches the nearest point $Q$ whose $y$-coordinate is $-8$, given by the numerical value of the difference in the $x$-coordinates of $P$ and $Q$.

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