For the function $f:[1, \infty) \rightarrow[1, \infty)$ defined by $f(x)=(x-1)^4+1$, among the two statements: (I) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x)\right\}$ contains exactly two elements, and (II) The set $\mathrm{S}=\left\{x \in[1, \infty): f(x)=f^{-1}(x+1)\right\}$ is an empty set,