A factory produces three types of sugar, types $X$, $Y$, and $Z$, from three types of syrup, labelled $A, B$, and C. The following table contains the number of litres of syrup necessary to make each kilogram of sugar.

\begin{tabular}{c|ccc} & $\mathrm{X}$ & $\mathrm{Y}$ & $\mathrm{Z}$ \ \hline $\mathrm{A}$ & 3 & 2 & 1 \ $\mathrm{~B}$ & 2 & 3 & 2 \ $\mathrm{C}$ & 4 & 1 & 2 \end{tabular}

For instance, one kilogram of type $\mathrm{X}$ sugar requires 3 litres of $\mathrm{A}, 2$ litres of $\mathrm{B}$, and 4 litres of C. The factory can sell each type of sugar for one pound per kilogram. Assume that the factory owner can use no more than 44 litres of $\mathrm{A}$ and 51 litres of $\mathrm{B}$, but is required by law to use at least 12 litres of C. If her goal is to maximize profit, how many kilograms of each type of sugar should the factory produce?