How am I supposed to do that?

Algebra Level 5

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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