Diagonalization & Matrix Powers · Hard problem

A product is manufactured in three plants; every month each unit of output moves between plants according to the transition matrix $P = \begin{pmatrix}\tfrac12 & \tfrac14 & 0\\ \tfrac12 & \tfrac12 & \tfrac12\\0 & \tfrac14 & \tfrac12\end{pmatrix}$ (column $j$ gives the fractions of plant $j$'s output that end up at each plant next month). Diagonalizing $P$ shows the distribution converges to the eigenvector for eigenvalue $1$. Find the long-run fraction of output at Plant 1.