\[\Large\begin{cases} \log_x w & = & 24 \\ \log_y w & = & 40 \\ \log_{xyz} w & = & 12 \end{cases}\]

If \(x,y,z\) are real numbers greater than 1 and \(w\) is a positive number satisfying the system above, then find the value of \(\left (\log_w z \right)^{-1}\).

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