Revival Of This Problem. Part 2

Geometry Level 4

\[\begin {cases} a^{2} + ab + b^{2} = 2 \\ b^{2} + bc + c^{2} = 1 \\ c^{2} + ca + a^{2} = 3 \end {cases}\]

Let \(a,b,c\) be all positive numbers satisfying the system of equations above. If \( ab + bc + ca = \cfrac{ x \sqrt{y} } { z} \), where \( x\) and \(z\) are positive coprime integers, and \(y\) is a squarefree integers, then find \( x + y + z \).

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