KöMaL 2026 A926

Let $A=\{[\frac{3+\sqrt{5}}{2}\cdot n]\colon n=1,2,3,\ldots\}=\{2,5,7,10,13,15,\ldots\}.$ Two players, First and Second, play a game with a pile of $N$ coins according to the following rules. The players move alternately, with First moving first. On each move, the player on turn chooses a number $k\in A$ and removes $k$ coins from the pile. A player who is unable to make a legal move loses the game. Determine, in terms of $N$, which player has a winning strategy, and describe such a winning strategy.