In coordinate space, a particle starts at the point $(2,3,4)$ and ends at the point $(-1,-3,-3),$ along the line connecting the two points.  Along the way, the particle intersects the unit sphere centered at the origin at two points.  Then the distance between these two points can be expressed in the form $\frac{a}{\sqrt{b}},$ where $a$ and $b$ are positive integers, and $b$ is not divisible by the square of a prime.  Find $a + b.$

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