HMMT November 2016 E5

Let the sequence $\left\{a_{i}\right\}_{i=0}^{\infty}$ be defined by $a_{0}=\frac{1}{2}$ and $a_{n}=1+\left(a_{n-1}-1\right)^{2}$. Find the product

$$\prod_{i=0}^{\infty} a_{i}=a_{0} a_{1} a_{2} \ldots$$