Sequence and Series: JEE Main 2026 Mathematics question

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 $\min_{0<\theta<\frac\pi2}f(\theta)=\max_{0<\theta<\pi}g(\theta)$. 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 $\frac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.