2017 IMO Shortlist N1

The sequence $a_{0}, a_{1}, a_{2}, \ldots$ of positive integers satisfies $$a_{n+1}=\left\{\begin{array}{ll} \sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { is an integer } \\ a_{n}+3, & \text { otherwise } \end{array} \quad \text { for every } n \geq 0\right.$$ Determine all values of $a_{0}>1$ for which there is at least one number $a$ such that $a_{n}=a$ for infinitely many values of $n$. (South Africa)

正整数序列 $a_{0}, a_{1}, a_{2}, \ldots$ 满足 a_{n+1}=\left\{\begin{array}{ll} \sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { 是整数 } \\ a_{n}+3, & \text { 否则 } \end{array} \quad \text { 对于每个 } n \geq 0\right。 确定 $a_{0}>1$ 的所有值，其中至少有一个数字 $a$，使得 $a_{n}=a$ 对于无限多个 $n$ 值。 (南非)