
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
Given an endless supply of white, blue and red cubes. In a circle arrange any  $N$  of them. The robot, standing in any place of the circle, goes clockwise and, until one cube remains, constantly repeats this operation: destroys the two closest cubes in front of him and puts a new one behind him a cube of the same color if the destroyed ones are the same, and the third color if the destroyed two are different colors. 
We will call the arrangement of the cubes *good* if the color of the cube remaining at the very end does not depends on where the robot started. We call  $N$  *successful* if for any choice of  $N$  cubes all their arrangements are good. Find all successful  $N$ .

I. Bogdanov

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