Trigonometric warping

This question is flagged because I believe that $\alpha, \beta, \gamma$ should be angles in a triangle.

If $|\cos{\alpha}\cos{\beta}\cos{\gamma}|=\frac{2}{3}$ . Then the maximum value of $\tan{\alpha}\tan{\beta}+\tan{\beta}\tan{\gamma}+\tan{\gamma}\tan{\alpha}$ is $X$ .

Find $24X$ .

Not Original