Working with Vieta

Algebra Level 5

Let a,b,ca,b,c be the real roots of the cubic equation 2t3+3t29t6=02t^3+3t^2-9t-6=0. Suppose that (abc+bca+cab)(bca+cab+abc)=mn\left|\left(\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}\right)\left(\dfrac{b-c}{a}+\dfrac{c-a}{b}+\dfrac{a-b}{c}\right)\right|=\dfrac{m}{n} for some relatively prime positive integers mm and nn. Find m+nm+n.

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