XOR Game

You are given an odd integer $N$ and two integer sequences $A_1, A_2, \ldots, A_N$ and $B_1, B_2, \ldots, B_N$. Your task is to reorder the elements of $B$, forming a new sequence $C_1, C_2, \ldots, C_N$ (i.e. choose a permutation $P_1, P_2, \ldots, P_N$ of the integers $1$ through $N$, where $C_i = B_{P_i}$ for each valid $i$), in such a way that the following condition holds: $(A_1 \oplus C_1