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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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The system of equations
\[ 4a^4+36a^2b^2+9b^4=20a^2+30b^2-1\]
\[2a^3b+3ab^3=5ab\]
has \(n\) nonnegative (meaning both \(a\) and \(b\) are nonnegative) real solutions. Let these solutions be \((a_1,b_1),(a_2,b_2),\cdots (a_n,b_n)\). The value of the sum
\[\sum_{i=1}^n a_i+b_i\]
can be expressed in the form \(\dfrac{a+b\sqrt{c}}{d}\) where \(\gcd (a,b,d)=1\) and \(c\) is squarefree. Find \(a+b+c+d\).

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