Eccoci.

Let $B(n)$ be the smallest number larger than $n$ that can be formed by rearranging digits of $n$, or $0$ if no such number exists. For example, $B(245)=254$ and $B(542)=0$.

Define $a_0=0$ and $a_n=a_{n-1}^2+2$ for $n>0$. Let ${\displaystyle U(N)=\sum _{n=1}^{N}B(a_n)}$. You are given $U(10)\equiv 543870437(\mathrm{mod}{10}^9+7)$.

Find $U({10}^{16})$. Give your answer modulo ${10}^9+7$.