Solve the following system for real \(x\) and \(y\) \[5x\left ( 1+\dfrac{1}{x^2+y^2}\right )=12\] \[5y\left ( 1-\dfrac{1}{x^2+y^2}\right )=4\]

**Note**

- Enter your answer as the sum of all such values of \(x\) and \(y\).
- This question once appeared in RMO.

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