Minimising can be Difficult!

Algebra Level 5

For real numbers \(x,y,z\), subject to \(x+y+z>6\) and

\[x^2+y^2+z^2-xyz=(x+y)(y+z)(z+x)-8(xy+yz+zx), \]

what is the minimum value of \(xy+yz+zx\)?

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