
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 \( n \) be an integer greater than 1. In a circular arrangement of \( n \) lamps \( L_0, L_1, \ldots, L_{n-1} \), each of which can be either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, Step 1, Step 2, ...

If \( L_{j-1} \) (with \( j \) taken mod \( n \)) is ON, then Step \( j \) changes the status of \( L_j \) (it goes from ON to OFF or from OFF to ON) but does not change the status of any of the other lamps.

(a) There is a positive integer \( M(n) \) such that after \( M(n) \) steps, all lamps are ON again.

(b) If \( n \) has the form \( 2^k \), then all lamps are ON after \( n^2 - 1 \) steps.

(c) If \( n \) has the form \( 2^k + 1 \), then all lamps are ON after \( n^2 - n + 1 \) steps.

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