Two disjoint closed classes give infinitely many stationary distributions

Consider a Markov chain on $S=\{1,2,3,4,5\}$ with transition probability matrix

$$P=\begin{pmatrix}3/4&0&1/4&0&0\\1/2&0&0&1/2&0\\1/2&0&1/2&0&0\\ 0&0&0&2/3&1/3\\0&0&0&1/3&2/3\end{pmatrix}.$$

Then: