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Algebra Level 4

Given that one of the root of \(f(x) = x^5+9x^4+7x^3+16x^2+6x+7\) can be expressed as:

\[-a-7\sqrt[3]{\dfrac{2}{b-3\sqrt{c}}} - \sqrt[3]{\dfrac{b-3\sqrt{c}}{2}}\ \]

for positive integers \(a,b\) and \(c\). Calculate \(a+b+c+1\).

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