Let $T = \{ 1, 2, 3, \dots, 14, 15 \}$. A subset $S$ of $T$ is defined as \textit{handy} if the sum of all the elements of $S$ is a multiple of $5$. For example, the empty set is handy (because its sum is $0$) and $T$ itself is handy (because its sum is $120$). Compute the number of handy subsets of $T$.

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