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 $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is equal to $\_\_\_\_$ .