Prism Volume from a Dividing Plane

Let $\mathscr{R}$ be a right rectangular prism with vertices $A_1, A_2, A_3, A_4, B_1, B_2, B_3, B_4$, where $A_1A_2A_3A_4$ and $B_1B_2B_3B_4$ are two parallel rectangular faces, with $A_1A_2 = B_1B_2 = 3$, $A_2A_3 = B_2B_3 = 7$, and $\overline{A_1B_1}$, $A_2B_2$, $A_3B_3$, and $A_4B_4$ are mutually parallel edges of $\mathscr{R}$. Suppose that a plane intersects segments $\overline{A_1B_1}$, $A_2B_2$, $A_3B_3$, and $A_4B_4$ at $P_1$, $P_2$, $P_3$, and $P_4$, respectively, dividing $\mathscr{R}$ into two solids, each with volume exactly $\frac{1}{2}$ that of $\mathscr{R}$. If three of the lengths $A_1P_1$, $A_2P_2$, $A_3P_3$, and $A_4P_4$ are 3, 4, and 6 in some order, then find the sum of all possible values of the volume of $\mathscr{R}$.