smc

We multiply all given expressions to get: $$(1)xyz+x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{xyz}=\frac{28}{3}$$ Adding all the given expressions gives that $$(2)x+y+z+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=4+\frac{7}{3}+1=\frac{22}{3}$$ We subtract $(2)$ from $(1)$ to get that $xyz+\frac{1}{xyz}=2$. Hence, by inspection, $\boxed{1}$.