Such Big Denominator you got there!

Algebra Level 3

We have \(3\) non-negative real numbers \({a,b,c}\) such that \(abc=1\).

When the maximum value of the expression

\[\dfrac{2}{2ab+2ac+2bc+a^2+b^2+c^2} \]

is expressed in the form \(\dfrac{x}{y}\) for coprime positive integers \(x\) and \(y\)

What is the value of \(x+y\)?

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