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 $\min\limits_{0<\theta<\frac{\pi}{2}} f(\theta)=\max\limits_{0<\theta<\pi} g(\theta)$. 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 $\_\_\_\_$ .

If $\sum\limits_{k=1}^{n} a_k = 6 n^3$, then $\sum\limits_{k=1}^{6} \left( \frac{a_{k+1} - a_k}{36} \right)^2$ is equal to ________.

If $\sum\limits_{r=1}^{25} \left( \frac{r}{r^4 + r^2 + 1} \right) = \frac{p}{q}$, where p and q are positive integers such that $\gcd(p, q) = 1$, then p + q is equal to ________.

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(a, b)$, then $a^2+b^2$ is equal to

$\_\_\_\_$ .