For the functions f(θ)=α tan^2 θ+β cot^2 θ, and g(θ)=α sin^2 θ+β cos^2 θ, α>β>0, let min_0<θ<(π)/2 f(θ)= max_0<θ<π g(θ). If the first term of a G.P.…

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