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

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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