
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
We want to place $n$ points on the surface of a sphere such that they "do not disturb each other." By this, following the idea of the Dutch mathematician L. B. Van den Waerden, we mean that the distance between any two placed points is at least 1 unit. What is the diameter of the sphere required for $n = 2, 3, 4, 5, 6,$ and $8$?

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