AMC 12 2022a 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 $p+q+r$?

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