Inequalities are Hard

Let  x, y, and z be real numbers such that \(x + y + z = 0\) and \(xyz = 2\).

 

 

Find the maximum value of \(x^3 y + y^3 z + z^3 x\).  

 

 

I honestly do not know where to start this problem because of this reasoning here:

 

 

Basically, we can deduce that one of the variables must be positive, and the other two negative, and because of this, the QM-AM-GM-HM inequalities have reduced functionality.  

 

Perhaps the Cauchy-Schwarz can be of use here?

 

 

Thanks for any help!