These Are Not The Solutions You Are Looking For

Let all of the solutions of non-negative integers to the equation

\[ 2x^{2}+x=3y^{2}+y \]

be \((x_{1},y_{1}),(x_{2},y_{2}),....\), where \(x_{1}+y_{1}<x_{2}+y_{2}<.....\). Find the value of \(\sum_{i=1}^{4}(x_{i}+y_{i})\).

Image credit: InSapphoWeTrust 2012
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