Math puzzle: The homesick rover

Far away, on a large, rocky exoplanet, a rover from Earth has just arrived on a mission of exploration. But the poor thing doesn’t want to explore. The moment it begins moving, it grows homesick and wants nothing more than to return to the very spot where it began. Unfortunately, it has received very explicit instructions. The first day, it must travel one kilometer forward in a straight line, then turn 90 degrees. The second day, it must travel two kilometers forward, then turn 90 degrees. The third day, it must travel three kilometers forward, then turn 90 degrees. The fourth day, it must travel four kilometers forward, then

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