BALKAN 2018 Problem 2

Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n=1,2, \ldots)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^{n}$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$ ), but have not taken exactly the same route within that time. Determine all possible values of $q$.