Given that $a$, $b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, determine the minimum possible positive value of $\frac{1}{a} - \frac{1}{b} - \frac{1}{c}$. This value can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

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