Mathematics Question from JEE Main 2026 · 22 January, Shift 2

Let $\cos(\alpha + \beta) = -\dfrac{1}{10}$ and $\sin(\alpha - \beta) = \dfrac{3}{8}$, where $0 < \alpha < \dfrac{\pi}{3}$ and $0 < \beta < \dfrac{\pi}{4}$.

If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, \quad r, s \in \mathbb{N}$, then the value of $r + s$ is _____.