What's C, G, and N?

Algebra Level 4

Let \(C\) be the answer to this problem.
Let \(G\) be the answer to this problem.
Let \(N\) be the answer to this problem.

Let \(m\) be the largest root of \(x^3 + Cx^2 + Gx - N = 0\). \(m\) can be written as \(\sqrt{a} - \sqrt{b}\), where \(a,b\) are positive integers. What is the value of \(a+b\)?

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