The weight of $\frac{1}{4}$ of a large cake together with $2\frac{1}{2}$ cups of mixed nuts is the same weight as $\frac{2}{3}$ of a large cake together with $\frac{1}{2}$ cup of mixed nuts. A cup of mixed nuts weighs $\frac{1}{3}$ of a pound. What is the weight, in pounds, of a large cake?