Example 4. A cubic water tank (Figure 6), with an edge length of $1 \mathrm{dm}$, has small holes $P, Q, R$ on edges $A A_{1}, B B_{1}, A_{1} L^{*}$, respectively, and $A_{1} P=A_{1} R=\frac{1}{3} \mathrm{dm}, B_{1} \dot{Q}=\frac{1}{4} \mathrm{dm}$. How much water can the tank hold at most? (Pipes can be placed arbitrarily)
Slightly explained: Let points $P, Q, N$ intersect $B_{1} C_{1}$ at S. $\because$ plane $A D D_{1} A_{1} / /$ plane $B C C_{1} B_{1}, \quad \therefore P R$ $/ / Q S$. The maximum volume that can be filled is the difference between the volume of the cube and the volume of the frustum of a triangular pyramid $B_{1} Q S-A_{1} P R$ (which can be proven to be a frustum), and this volume $=1-\frac{1}{3}\left(\frac{1}{18}+\frac{1}{24}+\frac{1}{32}\right)=1-\frac{37}{864}=\frac{827}{864}\left(\mathrm{dm}^{3}\right)$.