BALTIC-WAY 2016 Problem 10

Let $a_{0,1}, a_{0,2}, \ldots, a_{0,2016}$ be positive real numbers. For $n \geq 0$ and $1 \leq k<2016$ set

$$a_{n+1, k}=a_{n, k}+\frac{1}{2 a_{n, k+1}} \text { and } a_{n+1,2016}=a_{n, 2016}+\frac{1}{2 a_{n, 1}} .$$

Show that $\max _{1 \leq k \leq 2016} a_{2016, k}>44$.