Nested radicals

Algebra Level 5

Let R\mathcal{R} be the set of all the 210=10242^{10}=1024 real numbers of the form: ±2±2±2±±2,\pm\sqrt{2\pm\sqrt{2\pm \sqrt{ 2\pm \cdots \pm \sqrt{2}}}}, where root signs appear exactly 1010 times. If rR125+r=ab,\sum_{r\in \mathcal{R}}\frac{1}{25+r}=\frac{a}{b}, where aa and bb are coprime positive integers. What are the last three digits of a+ba+b?

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