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.