1980 AHSME Problem 13

A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to (1,0). Then it makes a 90^\circ counterclockwise and travels \frac 12 a unit to \left(1, \frac 12 \right). If it continues in this fashion, each time making a 90^\circ degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?

\text{(A)} \ \left(\frac 23, \frac 23 \right) \qquad \text{(B)} \ \left( \frac 45, \frac 25 \right) \qquad \text{(C)} \ \left( \frac 23, \frac 45 \right) \qquad \text{(D)} \ \left(\frac 23, \frac 13 \right) \qquad \text{(E)} \ \left(\frac 25, \frac 45 \right)

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.