Three A coins, three B coins and three C coins are online in the ABCABCABC order as shown in the drawing. You have to move them until they are in the AAABBBCCC order.
A movement is to support the index finger and the big toe on two coins that touch, and to move them to another place on the line.
During the move, the pair of coins can not be swapped, and after the move the pair must be touching some other coin of the line.
It may happen that some coins are isolated as a result of a play, but no pair can be taken to a position totally isolated from the rest.
It is not required that in the end all the coins are forming a compact line; there may be gaps.
Is it possible to do the rearrangement in four moves?