The master mind's problem

Algebra Level 4

\[\begin{cases} X^2+\dfrac{Y^{2}}{3}+XY=25 , \\ \dfrac{Y^{2}}{3}+Z^2=9 , \\ Z^2+ZX+X^2=16 \end{cases} \] If \(X,Y\) and \(Z\) are positive real numbers satisfying the system of equations above, find \( XY+2YZ+3ZX \).

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