A unitary divisor $d$ of a number $n$ is a divisor of $n$ that has the property $\gcd(d, n/d) = 1$.
The unitary divisors of $4! = 24$ are $1, 3, 8$ and $24$.
The sum of their squares is $1^2 + 3^2 + 8^2 + 24^2 = 650$.
Let $S(n)$ represent the sum of the squares of the unitary divisors of $n$. Thus $S(4!)=650$.
Find $S(100\,000\,000!)$ modulo $1\,000\,000\,009$.
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