Baby and the Couch

A baby is learning to walk with the assistance of its living room couch. The baby starts at the couch and at each time step makes a decision:

• take a step forward with probability $0.2$,
• stay in place with probability $0.5$,
• take a step back (towards the couch) with probability $0.3$.

The baby can never go behind the couch. Therefore, when the baby is at the couch, it moves forward with probability $0.2$ and stays at the couch with probability $0.8$.

If you observe the baby for an extremely long time, what proportion of the time is the baby at the couch?