Markov Chains & Long-Run Behavior · Superchallenge problem

Three files $a$, $b$, $c$ are kept in a list. At each step one file is requested, independently of the past: $a$ with probability $1/2$, $b$ with probability $1/3$, $c$ with probability $1/6$. The requested file is moved to the front of the list, and the other two keep their relative order. In the long run, what is the probability that the list reads $c, b, a$ from front to back? Express your answer as a common fraction.