OBM 2014 Problem 3

Let $N$ be an integer, $N>2$ . Arnold and Bernold play the following game: there are initially $N$ tokens on a pile. Arnold plays first and removes $k$ tokens from the pile, $1\le k < N$ . Then Bernold removes $m$ tokens from the pile, $1\le m\le 2k$ and so on, that is, each player, on its turn, removes a number of tokens from the pile that is between $1$ and twice the number of tokens his opponent took last. The player that removes the last token wins.

For each value of $N$ , find which player has a winning strategy and describe it.