HMMT November 2024 E5

Let $f$ be a function on nonnegative integers such that $f(0)=0$ and

$$f(3 n+2)=f(3 n+1)=f(3 n)+1=3 f(n)+1$$

for all integers $n \geq 0$. Compute the sum of all nonnegative integers $m$ such that $f(m)=13$.