Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.

\[\begin{cases} 4\dfrac{\sqrt{x^{2} + 1}}{x} = 5\dfrac{\sqrt{y^{2} + 1}}{y}=6\dfrac{\sqrt{z^{2} + 1}}{z} \\ xyz = x + y + z\end{cases}\]

If \(x = \dfrac{\sqrt{a}}{b} , y = c\dfrac{\sqrt{d}}{e} , z = f\sqrt{g}\) satisfy the above system of equations, then find \( (a + b + \ldots + g )\).

- The expression is in simplest terms, meaning that all of the variables are positive integers, \( a, d, g \) are square free and \(c, e \) are coprime.

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