Expected Value, Linearity & Advanced Probability · Superchallenge problem

A fair coin is flipped until the first head appears; let $N$ be the number of flips (so $N=1$ with probability $\tfrac12$, $N=2$ with probability $\tfrac14$, and so on). Then a fair six-sided die is rolled $N$ times. What is the expected number of different faces that appear on the die? Express your answer as a common fraction.