Probability problem #4957

A 2-player game is played on a $2 \times n$ grid of unit squares having $2$ rows and $n$ columns. At the start of the game, a token is placed on the top-left square. On a turn, a player is allowed to move the token one square to the right, two squares to the right, one square up, or one square down, provided that the token remains on the grid and does not move to a square that it has already been in. A player who is unable to move the token loses the game. For how many values of $n$ with $1 \leq n \leq 999$ does the first player have a winning strategy?