If the sum of infinite terms of the given series ${\cot }^{-1}\left(9+\frac{{\displaystyle 2}}{{\displaystyle 9}}\right)+{\cot }^{-1}\left(9+\frac{{\displaystyle 6}}{{\displaystyle 9}}\right)+{\cot }^{-1}\left(9+\frac{{\displaystyle 12}}{{\displaystyle 9}}\right)+...={\tan }^{-1}\left(\frac{{\displaystyle \mathrm{a}}}{{\displaystyle \mathrm{b}}}\right)$where a and b are relatively prime, then which of the following is/are true?

$\mathrm{a}>\mathrm{b}$

${\mathrm{a}}^3+{\mathrm{b}}^3=9$

${\sec }^{-1}\frac{\mathrm{a}}{2}\le {\mathrm{cosec}}^{-1}\left(\mathrm{b}\right)$

Mean deviation about mean of the data set containing all the integral elements lying in between$\left(2023+a\right)$and $\left(2023+b\right)$is one of the given elements of the given data set