A jumping leprechaun is a special chess piece that lives on an infinite square grid. On the first turn it moves one cell horizontally (left or right) and two cells vertically (up or down). On the $n$-th turn it moves $n$ cells horizontally and $n+1$ cells vertically. Both moves need to happen in a turn. Can the jumping leprechaun come back to its starting location? What is the least number of turns required for that?