How did I not see that..

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Find the sum of all possible values of \(\theta \in [-\pi, \pi]\) such that \[ \sin^3\left(\dfrac{1}{2}\theta\right) + \sin^3\left(\dfrac{1}{3}\theta\right) + \sin\left(\dfrac{1}{6}\theta\right) + \sin\left(\dfrac{1}{12}\theta\right) = 0 \]

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