
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 $(3, 4, 5)$ and moves to the point $(-2, -4, -6)$, along the line connecting these two points. As the particle progresses, it intersects with a sphere of radius 2 centered at the origin at two points. Calculate the distance between these two points, which is expressed in the form $\frac{a}{\sqrt{b}}$, where $a$ and $b$ are positive integers with $b$ not divisible by the square of a prime. Determine the sum $a + b$.

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