Not the Same
Two players are playing Not the Same. They alternate turns removing stones from a pile with \(N>0\) stones. On the first turn, the first player removes \(1\), \(2\), or \(3\) stones. On each subsequent turn, the player removes \(1\), \(2\), or \(3\) stones from the pile, but they cannot remove the same number of stones as the previous player removed in the previous turn. If a player cannot go, they forfeit their turn. The player who removes the last stone wins.
For how many positive integer initial pile sizes \(N\le 1000\) does the first player have a winning strategy?
Details and Assumption
If a player forfeits their turn, on the next turn the next player can remove 1, 2, or 3 stones.
A player cannot remove more stones than are left in the pile.