XOR Paths

The map of the Airship is an undirected graph. The undirected graph is constructed as follows. There are $2N$ vertices which are numbered from $1$ to $2N$, each having an associated value $A_i$. $A_{N + 1}, A_{N + 2}, \ldots, A_{2N}$ contain no duplicate elements, while $A_1, A_2, \ldots, A_N$ may contain duplicate elements, as well as elements in common with $A_{N + 1}, A_{N + 2}, \ldots, A_{2N}$