
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
Some  $n>2$  lamps are cyclically connected: lamp  $1$  with lamp  $2$ , ..., lamp  $k$  with lamp  $k+1$ ,..., lamp  $n-1$  with lamp  $n$ , lamp  $n$  with lamp  $1$ . At the beginning all lamps are off. When one pushes the switch of a lamp, that lamp and the two ones connected to it change status (from off to on, or vice versa). Determine the number of configurations of lamps reachable from the initial one, through some set of switches being pushed.

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