Three Variables, One Equation!

Algebra Level 4

\[\large{x+y+z+ \dfrac3{x-1} + \dfrac3{y-1} +\dfrac3{z-1} = 2 \left( \sqrt{x+2} + \sqrt{y+2} + \sqrt{z+2} \right)}\]

If the sum of all real numbers \(x,y,z \geq 1\) satisfying the above equality can be expressed as:

\[\dfrac{A}{B} \left(C+\sqrt{D}\right)\]

where \(A,B,C,D\) are positive integers, with \(\gcd(A,B)=1\) and \(D\) has no square factors, find the value of \(A+B+C+D\)?

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