Markov Chains & Long-Run Behavior · Challenge problem

A lamp always holds one bulb, whose age in days is $1$, $2$, $3$ or $4$. A bulb of age $k$ burns out at the end of the day with probability $k/4$ (so a bulb of age $4$ always burns out); a burnt-out bulb is replaced overnight by a new bulb, of age $1$ the next day, and otherwise the bulb's age goes up by $1$. In the long run, on what fraction of days does the bulb burn out? Express your answer as a common fraction.