Markov Chains & Long-Run Behavior · Hard problem

A factory's assembly line is each day Running, Slow or Stopped. From Running: stays Running with probability $4/5$, goes Slow with probability $3/20$, Stopped with probability $1/20$. From Slow: Running with probability $2/5$, stays Slow with probability $1/2$, Stopped with probability $1/10$. From Stopped: Running with probability $3/10$, Slow with probability $1/5$, stays Stopped with probability $1/2$. A Slow day costs $\$200$ in lost output and a Stopped day costs $\$500$; a Running day costs nothing. In the long run, what is the expected cost per day? Express your answer as a common fraction.