A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin. (a) If $\theta = 90^o$, determine all locations where the point could end up. (b) If $\theta = 45^o$, prove that for every location $ L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (c) Determine all rational numbers $\theta$ such that for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L$. (d) Prove that when $\theta$ is irrational, for every location $L$ in the coordinate plane and every positive number $\varepsilon$, there is a sequence of moves after which the point has distance less than $\varepsilon$ from $L.$