Professor Guilherme created three strange machines. Machine $A$ transforms a cat into a dog with a probability of $\frac{1}{3}$. Machine $B$ transforms a cat into a dog with a probability of $\frac{2}{5}$. Machine $C$ transforms a cat into a dog with a probability of $\frac{1}{4}$. And if the animal is a dog, none of the machines will perform any transformation.

Professor Guilherme placed a cat in machine $A$, then placed the resulting animal from machine $A$ in machine $B$, and finally placed the resulting animal from machine $B$ in machine $C$. What is the probability that a dog came out of machine $C$?