Switch

Logic Level 3

There is a circle of \(n\) light bulbs with a switch next to each of them. Each switch can be flipped between two positions, thereby toggling the on/off states of three lights: its own and the two lights adjacent to it. Initially, all the lights are off.

Let the minimum number of flips needed to turn on all the \(n=12\) and \(n=13\) light bulbs be \(a\) and \(b\), respectively. Then what is the value of \(a+b\)?

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