Weird sum

Algebra Level 4

Let \[f (a, b, c)=\frac {1}{1+a+ab+abc}\] If \(w, x, y, z\) are real numbers such that \(wxyz=1\), let \(M\) and \(m\) be the maximum and minimum values of \[f (w, x, y)+f (x, y, z)+f (y, z, w)+f (z, w, x)\] respectively. Find \(M-m\).

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