Justine has two fair dice, one with sides labeled $1, 2, \ldots, m$ and one with sides labeled $1, 2, \ldots, n.$ She rolls both dice once. If $\frac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$.

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