
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
Let  $V=[(x,y,z)|0\le x,y,z\le 2008]$  be a set of points in a 3-D space.
If the distance between two points is either  $1, \sqrt{2}, 2$ , we color the two points differently.
How many colors are needed to color all points in  $V$ ?

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