
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 $(1,2,2)$ and ends at the point $(-2,-2,-1),$ moving along the line connecting these two points. During its journey, the particle intersects a sphere with radius 2 centered at the origin at two points. Calculate the distance between these two points of intersection, which can be expressed in the form $\frac{a}{\sqrt{b}}$, where $a$ and $b$ are positive integers and $b$ is square-free. Determine the value of $a+b$.

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