Minimize Samuel's fraction

Algebra Level 4

What is the minimum value of (x2+x+1)(y2+y+1)(z2+z+1)(t2+t+1)xyzt \left| \frac{(x^{2} + x + 1)(y^{2} + y + 1)(z^{2} + z + 1)(t^{2} + t + 1)}{xyzt} \right| as x,y,z,tx,y,z,t range over all non-zero real numbers?

This problem is posed by Samuel Q.

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