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 $1\le a\le b\le c<a+b\le n$.

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.