Let $S(n)$ be the sum of all positive integers $m$ not exceeding $n$ having the following property:
$a^{m+4}\equiv a(\mathrm{mod}m)$ for all integers $a$.

The values of $m\le 100$ that satisfy this property are $1,2,3,5$ and $21$, thus $S(100)=1+2+3+5+21=32$.
You are given $S({10}^6)=22868117$.

Find $S({10}^{12})$.