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 \(a\) 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 the value of \(p+q+r\)?

Let's think step by step and output the final answer within \boxed{}.
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