Consider the function on three variables defined as \(F(x,y,z) = (xy, yz, zx)\). Let \(C\) be the twisted cubic polynomial curve given by \(\mathbf{r}(t) = (t,t^2,t^3)\) as \(t\) moves from 0 to 1. What is the value of \[ N = \int_C F \cdot dr ? \]

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