The absolute value of difference of the solutions of the equation $\sum _{m=1}^{6}Co\sec \left[\theta +\frac{\left(m-1\right)\pi }{4}\right]Co\sec \left[\theta +\frac{m\pi }{4}\right]=4\sqrt{2}$  where  $\theta \in \left[0,\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right]$ is _____
 

$\sum _{m=1}^6\frac{\sin [(\theta +m\frac{\pi }{4})-(\theta +\frac{(m-1)\pi }{4})]}{\sin \left(\theta +\frac{(m-1)\pi }{4}\sin (\theta +\frac{m\pi }{4}\right)}=4$

$\Rightarrow \sum _{m=1}^6\{\cot (\theta +\frac{(m-1)\pi }{4})-\cot (\theta +\frac{m\pi }{4})\}=4$

$\Rightarrow \cot \theta -\cot (\theta +\raisebox{1ex}{$3\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.)=4$

$\Rightarrow \cot \theta +\tan \theta =4$

$\Rightarrow \tan \theta =2\pm \sqrt{3}$

$\Rightarrow \theta =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$12$}\right.,\raisebox{1ex}{$5\pi $}\!\left/ \!\raisebox{-1ex}{$12$}\right.n(\theta ,\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.)$

$\therefore \raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$12$}\right.-\frac{\pi }{12}=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.$