Arithmetic and Geometric Sequences Together

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Suppose \(x,y,z\) is a geometric sequence with a common ratio of \(r\) where \(r \neq 1\). If \(3x, 26y, 17z\) is an arithmetic sequence, then the sum of all possible values of \(r\) can be expressed as \(\dfrac{m}{n}\). Find \(m+2n\).

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