Solve this simple problem 'Simply'

Algebra Level 2

If, \[\frac{x+y}{2}=\frac{y+z}{3}=\frac{z+x}{4}\]Then \(x:y:z\) can be expressed as a:b:c where \(a, b\) & \(c\) are positive co-prime integers. What's \(a+b+c\)? (\(x, y, z\) are positive reals.)

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