A function $f$ is defined on integers such that:

- $f(n) = n + 3$ if $n$ is odd.
- $f(n) = \frac{n}{2}$ if $n$ is even.

If $k$ is an odd integer, determine the values for which $f(f(f(k))) = k$.

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