
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 parabola $y = x^2 - x - 6$ at a point $P$ whose $y$-coordinate is $6$. Find the horizontal distance traveled by the particle, defined as the numerical value of the difference in the $x$-coordinates of $P$ and $Q$, where $Q$ is the nearest point on the parabola with a $y$-coordinate of $-6$.

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