Let $A = \{1, 2, \ldots, 100\}$ and $f(k)$, where $k \in \mathbb{N}$, be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k) = 50$?

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