Eccoci.

Let $R(a, b, c)$ be the maximum area covered by three non-overlapping circles inside a triangle with edge lengths $a$, $b$ and $c$.

Let $S(n)$ be the average value of $R(a, b, c)$ over all integer triplets $(a, b, c)$ such that $1 \le a \le b \le c \lt a + b \le n$.

You are given $S(2) = R(1, 1, 1) \approx 0.31998$, $S(5) \approx 1.25899$.

Find $S(1803)$ rounded to $5$ decimal places behind the decimal point.