Example 2 (2004 National College Entrance Examination, Hunan Province) Three machines, A, B, and C, independently process the same type of part. It is known that the probability of machine A processing a first-class part while machine B does not is $\frac{1}{4}$, the probability of machine B processing a first-class part while machine C does not is $\frac{1}{12}$, and the probability of both machines A and C processing first-class parts is $\frac{2}{9}$.
(1) Find the probability that each of the three machines A, B, and C processes a first-class part;
(2) One part is selected from those processed by machines A, B, and C for inspection. Find the probability that at least one of them is a first-class part.