AHSME 1980 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? $\text{(A)} \ \left(\frac 23, \frac 23 \right) \text{(B)} \ \left( \frac 45, \frac 25 \right) \text{(C)} \ \left( \frac 23, \frac 45 \right) \text{(D)} \ \left(\frac 23, \frac 13 \right) \text{(E)} \ \left(\frac 25, \frac 45 \right)$