Ostrich's Nested Radical

Algebra Level 5

Given that $$x$$ and $$y$$ are positive integers such that $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=\sqrt{y+\sqrt{y-\sqrt{y+\sqrt{y-\cdots}}}},$ find the sum of the first $$20$$ smallest possible values of $$x+y$$.

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