Give me a four, give me a two

Algebra Level 5

Let \(x,y,z\) be positive real numbers such that: \[\begin{align} xyz&=945 \\ x(y+1)+y(z+1)+z(x+1)&=385. \end{align} \] If the minimum possible value of \(z+\frac{y}{2}+\frac{x}{4}\) can be written as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?

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