
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
A particle travels from point $(1,1,1)$ to point $(4, 5, 6)$ in coordinate space. Determine where this path would intersect a sphere centered at the origin with a radius of 2, and find the distance between these two intersection points.

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