CONO-SUR 2016 Problem 3

There are $2016$ positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players $A$ and $B$ take turns making moves. Player $A$ has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.