For the functions $$f(\theta) = \alpha \tan^2\theta + \beta \cot^2\theta$$, and $$g(\theta) = \alpha \sin^2\theta + \beta \cos^2\theta$$, $$\alpha > \beta > 0$$, let $$\min_{0 < \theta < \frac{\pi}{2}} f(\theta) = \max_{0 < \theta < \pi} g(\theta)$$. If the first term of a G.P. is $$\left(\frac{\alpha}{2\beta}\right)$$, its common ratio is $$\left(\frac{2\beta}{\alpha}\right)$$ and the sum of its first 10 terms is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to _______.

$$\frac{\alpha \tan^2 \theta + \beta \cot^2 \theta}{2} \ge \sqrt{(\alpha \tan^2 \theta)(\beta \cot^2 \theta)}$$

$$\frac{f(\theta)}{2} \ge \sqrt{\alpha \beta} \implies f(\theta) \ge 2\sqrt{\alpha \beta}$$

$$\min_{0 < \theta < \frac{\pi}{2}} f(\theta) = 2\sqrt{\alpha \beta}$$

$$g(\theta) = \alpha \sin^2 \theta + \beta (1 - \sin^2 \theta)$$

$$g(\theta) = (\alpha - \beta)\sin^2 \theta + \beta$$

$$\max_{0 < \theta < \pi} g(0) = (\alpha - \beta)(1) + \beta = \alpha$$

$$2\sqrt{\alpha \beta} = \alpha$$

$$4\alpha \beta = \alpha^2$$

$$\alpha = 4\beta \implies \frac{\beta}{\alpha} = \frac{1}{4}$$

First term ($$a$$) = $$\frac{\alpha}{2\beta} = \frac{4\beta}{2\beta} = 2$$

Common ratio ($$r$$) = $$\frac{2\beta}{\alpha} = 2\left(\frac{1}{4}\right) = \frac{1}{2}$$

$$S_{10} = \frac{a(1 - r^{10})}{1 - r}$$

$$S_{10} = \frac{2\left(1 - \left(\frac{1}{2}\right)^{10}\right)}{1 - \frac{1}{2}} = \frac{2\left(1 - \frac{1}{1024}\right)}{\frac{1}{2}}$$

$$S_{10} = 4 \times \frac{1023}{1024} = \frac{1023}{256}$$

$$m = 1023, \quad n = 256$$

$$m + n = 1023 + 256 = 1279$$