
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 a coordinate space, a particle starts at the point $(1,2,1)$ and ends at the point $(4,0,-2)$, traveling along the line connecting the two points. Along the way, the particle intersects the sphere centered at $(1,1,1)$ with radius 3 at two points. Compute the distance between these two intersection points.

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