
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 three-dimensional coordinate space, a particle starts at the point \((1, 1, 2)\) and ends at the point \((-2, -5, -3)\). Along the path, which is the line connecting the two points, the particle intersects both a unit sphere centered at the origin and a plane defined by the equation \(x + y + z = 0\). Calculate the distance between the two points where the particle intersects the unit sphere.

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