Question

An industrial chemical reservoir is emptied by three pumps, $P_1$, $P_2$, and $P_3$, operating independently at different constant rates. Working alone, $P_1$ can empty the full reservoir in $6$ hours, $P_2$ can empty it in $8$ hours, and $P_3$ can empty it in $12$ hours. Initially, the reservoir is completely full. First, $P_1$ and $P_2$ work together for $2$ hours. Then $P_1$ is turned off, and $P_3$ is turned on to work alongside $P_2$. Additionally, while $P_2$ and $P_3$ are working together, a defect causes liquid to leak out of the bottom of the reservoir at a constant rate equal to $\frac{1}{6}$ of the combined emptying rate of $P_2$ and $P_3$. How many additional hours will it take to completely empty the remaining liquid from the reservoir?