A function $f$ from the integers to the integers is defined as follows:
\[ f(n) = \begin{cases} n + 3 & \text{if } n \text{ is odd} \\
\frac{n}{2} & \text{if } n \text{ is even} \end{cases} \]
Suppose $k$ is odd and $f(f(f(k))) = 27$. Find the sum of the digits of $k$.

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