Cubics one step ahead of quadratics

Algebra Level 5

Suppose \(a\) and \(b\) are two positive real numbers such that the roots of the equation \(x^{3} - ax + b = 0\) are all real. Let \(t\) is a root of the equation with mininmal absolute value. If maximum value of \(t\) is of the form \(\dfrac{mb}{na}\). Find \(m + n\).

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