Sum This!

Algebra Level 3

\[ \Large{\frac1{\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}} + \frac1{\sqrt[3]{4}+\sqrt[3]{6}+\sqrt[3]{9}} + \frac1{\sqrt[3]{9}+\sqrt[3]{12}+\sqrt[3]{16}}} \]

If the expression above can be simplified to the form of \({\sqrt[3] a + b} \) for integers \(a\) and \(b\), find the value of \(a+b\).

Source: \(MA\Theta\) 1992.
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