Jonah is getting married to his girlfriend, Jorene. For the wedding, they are buying a custom $3$-layer cake, with each layer being a cylinder that is the same height but only $75\%$ of the diameter of the layer below. For example, if the base layer of the cake is $1$ foot tall and $1$ foot diameter, the layer on top of it would be $1$ foot tall and $9$ inches in diameter. Assume that Jonah and Jorene are paying for the cake at a constant rate per cubic inch.
If the base layer of the cake costs $\$128$, how much will the entire cake cost?