Russelle's triple tangent

Algebra Level 4

Let x,y,zx, y, z be the real roots of the cubic equation


and let ω=tan1x+tan1y+tan1z\omega = \tan^{-1} x+\tan^{-1} y+\tan^{-1} z. If tanω=ab\tan \omega = \frac{a}{b}, where aa and bb are positive coprime integers, what is the value of a+ba+b?

This problem is posed by Russelle G.


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