
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, $n$ lamps are arranged, with only one lamp lit at the start. At each step, you can change the state of three consecutive lamps. For which values of $n$ is it possible that after a certain number of steps, all the lamps are lit?

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