Let $f(n) = \frac{n}{3}$ if $n$ is divisible by $3$, and $f(n) = 4n - 10$ otherwise. Find the sum of all positive integers $c$ such that $f^5(c) = 2$. (Here, $f^5(x)$ means $f(f(f(f(f(x)))))$.)

Let's think step by step and output the final answer within \boxed{}.
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