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Algebra Level 5

Let \(x,y,z\) be positive reals such that \[x^2+y^2+z^2=1\] if \[axy+byz=\dfrac{\sqrt{a^2+b^2}}{2}\] for some positive real numbers \(a\) and \(b\), then the value of \(y\) can be expressed as

\[\large{\dfrac{A}{B\sqrt{C}} }\]

where \(A\) and \(B\) are co-prime positive integers and a square free natural number \(C\). Find \(A+B+C\).

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