Marginalizing a mixed initial distribution through one step of the transition matrix

Let $\{X_n\}_0^\infty$ be a Markov chain with $P[X_0=0]=P[X_0=1]= 1/4$ and $P[X_0=2]=1/2$. The t.p.m. is given by $P=\left(\begin{smallmatrix}1/2&1/3&1/6\\0&1/3&2/3\\1/2&0&1/2\end{smallmatrix}\right)$ (rows/columns indexed $0,1,2$). Then $P[X_1=1]$ is given by: