HMMT 2014 U29

Natalie has a copy of the unit interval $[0,1]$ that is colored white. She also has a black marker, and she colors the interval in the following manner: at each step, she selects a value $x \in[0,1]$ uniformly at random, and (a) If $x \leq \frac{1}{2}$ she colors the interval $\left[x, x+\frac{1}{2}\right]$ with her marker. (b) If $x>\frac{1}{2}$ she colors the intervals $[x, 1]$ and $\left[0, x-\frac{1}{2}\right]$ with her marker.

What is the expected value of the number of steps Natalie will need to color the entire interval black?