Abby and Brian play the following game: They first choose a positive integer $N$. Then they write numbers on a blackboard in turn. Abby starts by writing a 1. Thereafter, when one of them has written the number $n$, the other writes down either $n+1$ or $2 n$, provided that the number is not greater than $N$. The player who writes $N$ on the blackboard wins. (a) Determine which player has a winning strategy if $N=2011$. (b) Find the number of positive integers $N \leqslant 2011$ for which Brian has a winning strategy.