\[f(a)=\sqrt[3]{a+\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}+\sqrt[3]{a-\frac{a+1}{3}\sqrt{\frac{8a-1}{3}}}\]

For \(a\geq \dfrac18\), we define \(f(a)\) as above. Let \(x\) and \(y\) be the minimum and maximum values of \(f(a)\) respectively. Find \(100x-31y\).

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