For the function \( f(\theta)=\alpha \tan^2\theta+\beta \cot^2\theta \), and \( g(\theta)=\alpha \sin^2\theta+\beta \cos^2\theta, \ \alpha,\beta>0 \), let
If the first terms 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 ______.
