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Logic Level 2

There is a circle of nn 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=12n=12 and n=13n=13 light bulbs be aa and bb, respectively. Then what is the value of a+ba+b?

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