The function $f$ is defined for all positive integers as follows:
$$\begin{align*} f(1) &= 1\\ f(2n) &= 2f(n)\\ f(2n+1) &= 2n+1 + 2f(n)+\tfrac 1n f(n) \end{align*}$$It can be proven that $f(n)$ is integer for all values of $n$.
The function $S(n)$ is defined as $S(n) = \displaystyle \sum_{i=1}^n f(i) ^2$.
For example, $S(10)=1530$ and $S(10^2)=4798445$.
Find $S(10^{16})$. Give your answer modulo $1\,000\,000\,007$.
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