
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 $(1,2,3)$ and ends at the point $(0,-2,-2)$, along the line connecting these two points. The particle intersects a sphere of radius 2 centered at the origin at two points. Determine the distance between these two intersection points, which 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 $a + b$.

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