Given that the sum of all possible values of \(x\) that satisfy the equation

\[|3x-2|-|x+1|=x+2\]

can be expressed in the form \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, positive integers, find the value of \(m+n\).

This problem is part of the set Absolute Value Equations.

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