Appropriate Substitution

Algebra Level 3

\[ x^3 - 6x^2 + 12x = 8 \]

If \(a,b,c\) are the roots of the cubic equation above. And given that for coprime positive integers \(m,n\), we have:

\[ \left ( \frac 1 a + \frac 1 b + \frac 1 c \right )^2 = \frac m n \]

What is the value of \(16(m-n) \)?

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