Markov Chains & Long-Run Behavior · Challenge problem

A Markov chain on the states $0, 1, 2, 3$ moves as follows. From $0$ it moves to $1$ with probability $\tfrac12$ and otherwise stays at $0$. From $1$ it moves to $0$, stays at $1$, or moves to $2$, each with probability $\tfrac13$. From $2$ it moves to $1$ with probability $\tfrac12$, stays with probability $\tfrac14$, and moves to $3$ with probability $\tfrac14$. From $3$ it always moves to $2$. What is the steady-state probability of state $3$? Express your answer as a common fraction.