An irreducible chain of period 3: a unique stationary law, but no limit

Consider a Markov chain on $S=\{1,2,3,4\}$ with the transition probability matrix given by

$$P=\begin{bmatrix}0&0&0&1\\0&0&0&1\ \tfrac12&\tfrac12&0&0\\0&0&1&0\end{bmatrix}.$$

Then: