Markov Chains & Long-Run Behavior · Hard problem

A traffic study classifies commuters as Driving, taking Transit, or Biking. Of those who drove yesterday, $60\%$ drive again today, $20\%$ take transit, $20\%$ bike. Of those who took transit, $\tfrac16$ drive, $\tfrac23$ take transit again, $\tfrac16$ bike. Of those who biked, $15\%$ drive, $15\%$ take transit, $70\%$ bike again. If a commuter drove today, what is the probability that they will be biking two days from today? Express your answer as a common fraction.