The function $f$ is defined for all positive integers as follows:

• $f(1)=1$
• $f(3)=3$
• $f(2n)=f(n)$
• $f(4n+1)=2f(2n+1)-f(n)$
• $f(4n+3)=3f(2n+1)-2f(n)$

The function $S(n)$ is defined as $\sum _{i=1}^{n}f(i)$.

$S(8)=22$ and $S(100)=3604$.

Find $S(3^{37})$. Give the last $9$ digits of your answer.