Ostrich's Nested Radical

Algebra Level 5

Given that xx and yy are positive integers such that x+x+x+x+=y+yy+y,\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}=\sqrt{y+\sqrt{y-\sqrt{y+\sqrt{y-\cdots}}}}, find the sum of the first 2020 smallest possible values of x+yx+y.

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