smc

Let $w$ be the length of the shorter segment and $l$ be the length of the longer segment. We're given that $\frac{w}{l}=\frac{l}{w+l}$. Cross-multiplying, we find that $w^2+wl=l^2⟹w^2+wl-l^2=0$. Now we divide both sides by $l^2$ to get ${\left(\frac{w}{l}\right)}^2+\left(\frac{w}{l}\right)-1=0$. Therefore, $R^2+R-1=0$. From this, we have $R^2=-R+1$. Dividing both sides by $R$, we get $R=-1+\frac{1}{R}⟹R^{-1}=R+1$. Therefore, $R^2+R^{-1}=-R+1+R+1=2$. Finally, we have $$R^{[R^{(R^2+R^{-1})}+R^{-1}]}+R^{-1}=R^{[R^2+R^{-1}]}+R^{-1}=R^2+R^{-1}=\boxed{2}.$$