
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 \( n \geq 1 \) be a positive integer. Consider a row of \( n \) lamps. Initially, some lamps are on, and at each minute, lamps that have exactly one neighboring lamp on will turn on, and the others will turn off (lamps at the edges have only one neighbor).

For which integers \( n \) is it certain that no matter the initial configuration of the lamps, they will all be off after some time?

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