12. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, $E$ is the midpoint of $B C$. Find a point $P$ on edge $O C_{1}$ such that plane $A_{1} B_{1} P \perp$ plane $C_{1} D E$.
12. Establish a rectangular coordinate system as shown in the figure, then $A_{1}(1,0,1) 、 B_{1}(1,1,1) 、 C_{1}(0,1,1)$, and $E\left(\frac{1}{2}, 1,0\right)$. Let point $P(0,1, z)$, then $\vec{A}_{1} B_{1}=(0,1,0), \overrightarrow{C_{1} E}=\left(\frac{1}{2}, 0,-1\right)$, $\overrightarrow{B_{1}} \vec{P}=(-1,0, z-1)$.
Thus, $\overrightarrow{C_{1} E} \cdot \overrightarrow{A_{1} B_{1}}=0$.
From $\overrightarrow{C_{1} E} \cdot \overrightarrow{B_{1} P}=-\frac{1}{2}+0-(z-1)=0$, we get $z=\frac{1}{2}$.
Therefore, when $P$ is the midpoint of $C C_{1}$, the plane $A_{1} B_{1} P \perp$ plane $C_{1} D E$.