Switch

Logic Level 2

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 $a+b$ ?