Russelle's triple tangent

Algebra Level 4

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

$2u^3-799u^2-400u-1=0$

and let $\omega = \tan^{-1} x+\tan^{-1} y+\tan^{-1} z$ . If $\tan \omega = \frac{a}{b}$ , where $a$ and $b$ are positive coprime integers, what is the value of $a+b$ ?

This problem is posed by Russelle G.