2018 IMO Shortlist A4

Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers such that $a_{0}=0, a_{1}=1$, and for every $n \geq 2$ there exists $1 \leq k \leq n$ satisfying $$a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}$$ Find the maximal possible value of $a_{2018}-a_{2017}$. (Belgium)

设 $a_{0}, a_{1}, a_{2}, \ldots$ 为实数序列，使得 $a_{0}=0, a_{1}=1$，并且对于每个 $n \geq 2$ 存在 $1 \leq k \leq n$ 满足 $$a_{n}=\frac{a_{n-1}+\cdots+a_{n-k}}{k}$$ 找到可能的最大值$a_{2018}-a_{2017}$ 的值。 (比利时)