A particle is placed on the parabola $y = x^2 - x - 6$ at a point $P$ whose $y$-coordinate is $6$. It is allowed to roll along the parabola until it reaches the nearest point $Q$ whose $y$-coordinate is $-6$. Find the horizontal distance traveled by the particle, which is the numerical value of the difference in the $x$-coordinates of $P$ and $Q$.

Let's think step by step and output the final answer within \boxed{}.
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