An Algebraic Trigonometry!

Geometry Level 4

\[ \begin{cases} \tan(x)+\tan(y)+\tan(z)= 6-(\cot(x)+\cot(y)+\cot(z)) \\ \tan^2(x)+\tan^2(y)+\tan^2(z)= 6-(\cot^2(x)+\cot^2(y)+\cot^2(z) ) \\ \tan^3(x)+\tan^3(y)+\tan^3(z)= 6-(\cot^3(x)+\cot^3(y)+\cot^3(z) ) \\ \end{cases}\]

If \(x,y\) and \(z\) are real numbers that satisfy the three equations above. Find the value of the expression below.

\[ \left \lfloor \frac{\tan(x)}{\tan(y)} + \frac{\tan(y)}{\tan(z)} + \frac{\tan(z)}{\tan(x)} + 3\tan(x) \tan(y) \tan(z) \right \rfloor \]

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