$1.$ Consider two tanks $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash ).$ Suppose that $\backslash (\backslash$operatorname$\{$tank$\}$ x $\backslash )$ contains $100$ gallons of brine in which $100$ pounds of salt are dissolved, while tank $\backslash ($ y $\backslash )$ contains $100$ gallons of water. Water flows into $\backslash (\backslash$operatorname$\{$tank$\}$ x $\backslash )$ at the rate of $3$ gallons per minute, and the mixture flows from $\backslash ($ x $\backslash )$ into $\backslash ($ y $\backslash )$ at the rate of $4$ gallons per minute. From $\backslash ($ y $\backslash ),$ one gallon per minute is pumped back to $\backslash ($ x $\backslash ),$ while $3$ gallons per minute are removed from the system $($see figure below$).$ Find the maximum amount of salt in tank $\backslash ($ y $\backslash ).$