Nested radicals

Algebra Level 5

Let \(\mathcal{R}\) be the set of all the \(2^{10}=1024\) real numbers of the form: \[\pm\sqrt{2\pm\sqrt{2\pm \sqrt{ 2\pm \cdots \pm \sqrt{2}}}},\] where root signs appear exactly \(10\) times. If \[\sum_{r\in \mathcal{R}}\frac{1}{25+r}=\frac{a}{b},\] where \(a\) and \(b\) are coprime positive integers. What are the last three digits of \(a+b\)?

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