For all positive numbers \(x, y, z\), evaluate the product:
\[
(x+y+z)^{-1}(x^{-1}+y^{-1}+z^{-1})(xy+yz+xz)^{-1}[(xy)^{-1}+(yz)^{-1}+(xz)^{-1}]
\]
What is the simplified expression for this product? The simplified expression is in the form \(x^{-a}y^{-b}z^{-c}\). Please provide the value of \(a+b+c\).

Let's think step by step and output the final answer within \boxed{}.
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