Let \(x, y, z\) be positive real numbers such that: \(x^2+y^2+z^2+1=2(xy+yz+zx)\).

The minimum value of \(P=3x+2y+6z\) is \(\alpha\) iff \(x=\beta; y=\gamma; z=\delta\).

And \(\alpha+\beta+\gamma+\delta=\dfrac{m}{n}\) where \(m\) and \(n\) are co-prime positive integers.

What is the value of \(m+n\)?

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