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$\large \sum_{k=1}^{\infty} \dfrac{(-1)^{k+1} \sin(k)}{k^3}$

If the above sum can be represented as $\dfrac{\pi^a - b}{c}$ for positive integers $a$, $b$ and $c$, find the value of $a+b+c$.

Clarification: Angles are measured in radians.

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