Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3 + 18 + 3$, $18 + 3 + 3$, and $3 + 6 + 3 + 9 + 3$, but not $18 + 6$ or $24$.

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