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 $a+b$?