
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 infinite supply of white, blue, and red cubes. Any \(N\) of these cubes can be arranged in a circle. A robot, starting from any position in the circle, moves clockwise and repeatedly performs the following operation until only one cube remains: it destroys the two nearest cubes in front of it and places a new cube behind itself. The color of the new cube is the same if the destroyed cubes are of the same color, and of the third color if the destroyed cubes are of different colors. We call a cube arrangement good if the color of the final remaining cube does not depend on the robot's starting position. We call \(N\) lucky if, for any choice of \(N\) cubes, all their arrangements are good. Find all lucky \(N\).

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