Let $P=\left[\begin{array}{ccc}3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0\end{array}\right]$, where $\alpha \in R$. Suppose $Q=\left[q_{\text {ij }}\right]$ is a matrix satisfying $P Q=k l_3$ for some non-zero $k \in R$. If $q_{23}=-\frac{k}{8}$ and $|Q|=\frac{k^2}{2}$, then $\alpha^2+k^2$ is equal to $\_\_\_\_$ .