A number theory problem by Kian Chua

Consider positive integers solutions \((x,y,z,u)\) of the systems of equations

\[ \begin{cases} x+y=3(z+u), \\ x+z=4(y+u), \\ x+u=5(y+z). \\ \end{cases} \]

What is the smallest value that \(x\) can have?

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