Putnam 1992 B3

For any pair $(x, y)$ of real numbers, a sequence $(a_n(x,y))_{n\geq 0}$ is defined as follows:

$$\begin{aligned}a_0(x, y) &= x, \\ a_{n+1}(x, y) &= \frac{(a_n(x, y))^2 + y^2}{2}, \text{for $n \geq 0$}.\end{aligned}$$

Find the area of the region $$\{ (x, y) | (a_n(x, y))_{n \geq 0}\ \text{converges}\}.$$