A bacteria culture grows according to the equation $N(t) = N_0 \cdot 2^{t/3}$, where $N_0$ is the initial number of bacteria and $t$ is the time in hours. If the initial number of bacteria is $200$ and the culture grows to $1600$ bacteria, how many hours have passed? Express your answer in the form of a simplified radical expression. Let's think step by step and output the final answer within \boxed{}.
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