
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
On a circle, \( n \geq 3 \) points are marked. Each marked point is coloured red, green, or blue. In one step, one can erase two neighbouring marked points of different colours and mark a new point between the locations of the erased points with the third colour. In a final state, all marked points have the same colour which is called the colour of the final state. Find all \( n \) for which there exists an initial state of \( n \) marked points with one missing colour, from which one can reach a final state of any of the three colours by applying a suitable sequence of steps.

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