Successive reflection

Given 2012 points on the plane: $A_1, \dots, A_{2012}$, and a point $P$. Suppose $B_1, \dots, B_{2012}$ is a permutation of the $A_i$s, we determine the shadow of $P$ as follows:

Reflect $P$ with respect to $B_1$ to obtain $P_1$. Reflect $P_1$ with respect to $B_2$ to obtain $P_2$, and so on, to arrive with $P_{2012}$. We call this last point the shadow of $P$.

Obviously, depending on the permutation of $B_i$s, one may arrive at different shadows of $P$. Find the maximal numbers of shadows of $P$ over all possible permutations.