
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
In coordinate space, a particle starts at the point $(3,1,2)$ and ends at the point $(-2, -4, -5),$ moving along the line connecting these points. It intersects a sphere of radius 2 centered at the origin at two points. The distance between these two points can be expressed in the form $\frac{p}{\sqrt{q}},$ where $p$ and $q$ are positive integers, and $q$ is not divisible by the square of a prime.  Find $p + q.$

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