States 0, 1, 2 all communicate: the chain is irreducible

Consider the MC consisting of the three states 0, 1, 2 and having TPM

$$P=\begin{pmatrix}\tfrac12&\tfrac12&0\ \tfrac12&\tfrac14&\tfrac14\ 0&\tfrac13&\tfrac23\end{pmatrix}.$$

Which of the following is correct?