Suppose  $\alpha ,\beta ,\gamma$ are complex numbers satisfying the following system of the equations  $\alpha +\beta +\gamma =6,{\alpha }^3+{\beta }^3+{\gamma }^3=87$ and $(\alpha +1)(\beta +1)(\gamma +1)=33$ . If $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=k$ , then the value of  $\left[\frac{1}{k}\right]$  is (where [.] denotes greatest integer function)

Let $S_1=\alpha +\beta +\gamma$  and  $S_2=\alpha \beta +\beta \gamma +\gamma \alpha$ and  $S_3=\alpha \beta \gamma$  
$\therefore S_1=6$  and  $(\alpha +1)(\beta +1)(\gamma +1)=33$
 $\Rightarrow S_3+S_1+S_2=32$
$\Rightarrow S_2+S_3=26$   ………. (1)
Also  ${\alpha }^3+{\beta }^3+{\gamma }^3=87$
 $\Rightarrow S_1({S_1}^2-3S_2)+3S_3=87$
 $\Rightarrow 216+18S_2+S_3=87$
 $\Rightarrow 18S_2-3S_3=129$
$\Rightarrow 6S_2-S_3=43$   ………. (2)
Adding (1) and (2)
$\Rightarrow 7S_2=69\Rightarrow S_2=\frac{69}{7}$  and  $S_3=\frac{113}{7}$
 $\therefore \frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }=\frac{S_2}{S_3}=\frac{69}{113}$