Let $\{E_1, E_2, \dots, E_m\}$ be a collection of sets such that $E_i \subseteq X = \{1, 2, \dots, 100\}$, $E_i \neq X$, for $i = 1, 2, \dots, m$. It is known that every two elements of $X$ are contained together in exactly one $E_i$ for some $i$. Determine the minimum value of $m$.

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