Curious Condition

Algebra Level 5

Positive reals \(x,y,z\ge \dfrac{\sqrt{3}}{3}\) satisfy the condition \(xyz+x+y-z=0\). If \(kxyz-xy-yz-zx\ge 1\) is always true, the the minimum value of \(k\) can be expressed as \(\dfrac{a\sqrt{b}}{c}\) for positive integers \(a,b,c\) with \(a,c\) coprime and \(b\) square-free.

What is \(a+b+c\)?

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