Square Summation

Let $S(n)$ be the number of ways to write $n = a^2 + b^2$, where $(a,b)$ is a pair of integers. For example, $S(1) = 4$, as $1$ can be written in four ways:

$$1 = 0^2 + 1^2 = 1^2 + 0^2 = (-1)^2 + 0^2 = 0^2 + (-1)^2$$

Evaluate $\displaystyle \lim_{n \rightarrow \infty} \dfrac{1}{n}\sum_{i=1}^n S(n)$ to the nearest thousandth.