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Problem 924

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Larger Digit Permutation II

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 ${10}^9+7$.

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更大的数字重排（二）

令$B(n)$为通过重排$n$的数字所能形成的比$n$大的最小数，如果不存在这样的数则为$0$。例如，$B(245)=254$，$B(542)=0$。

定义$a_0=0$，且对于$n>0$有$a_n=a_{n-1}^2+2$。
令${\displaystyle U(N)=\sum _{n=1}^{N}B(a_n)}$。已知$U(10)\equiv 543870437(\mathrm{mod}{10}^9+7)$。

求$U({10}^{16})$，并对${10}^9+7$取余作为你的答案。

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