<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 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 $a + b.$ Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
59
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 $a + b.$
test