KöMaL 2016 B4804

With his sword received from the queen of the lakes, Sir Robin wants to free the town of Agnor from a dragon. The dragon has three heads and three tails. With one slash, he may cut off one head, two heads, one tail or two tails. If he cuts off one head, the dragon will immediately grow three heads in its place. If he cuts off two heads, then nothing will grow in its place. If he cuts off one tail, there will be two tails growing immediately. Finally, if he cuts off two tails then the dragon will grow a head. What is the minimum number of cuts needed to kill the dragon (that is, in order that the dragon has no heads or tails remaining at all)?