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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