2022 AMC 12A Problem 17

Suppose a is a real number such that the equation a\cdot(\sin{x}+\sin{(2x)}) = \sin{(3x)} has more than one solution in the interval (0, \pi). The set of all such a that can be written in the form (p,q) \cup (q,r), where p, q, and r are real numbers with p < q< r. What is p+q+r?

\textbf{(A) } {-}4 \qquad \textbf{(B) } {-}1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 4

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.