Trigonometric problem solving culminates in this chapter. Leave no side and no angle unmeasured!

In triangle \(ABC\), let \(a, b\) and \(c\) be the lengths of the legs of a triangle opposite vertices \(A, B, C \) respectively. Let \( \alpha, \beta, \gamma \) be the values of the respective angles. What is the value of

\[ a ( \sin \beta - \sin \gamma) + b ( \sin \gamma - \sin \alpha ) + c ( \sin \alpha - \sin \beta )? \]

Triangle \(ABC\) satisfies \(2\sin B=\sin A+\sin C\). If \(a+c=30\), what is the value of \(b\)?

**Details and assumptions**

\(a\), \(b\) and \(c\) are the lengths of the sides opposite to the vertices \(A\), \(B\) and \(C\), respectively.

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