Optimizing the "Cool Function"

Algebra Level 4

Let the Cool Function $f$ be defined from the positive reals by $f(x,y,z,t) = \frac{(x^{2} + 2x + 1)(y^{2} + 2y + 1)(z^{2} + z + 1)(t^{2} + t + 1)}{xyzt}$ Find the minimum value of $f$ .