HMMT 2020 U27

Let $\left\{a_{i}\right\}_{i \geq 0}$ be a sequence of real numbers defined by

$$a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}$$

for $n \geq 0$. Determine the largest value for $a_{0}$ such that $\left\{a_{i}\right\}_{i \geq 0}$ is bounded.