Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we define $f(p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f(p)$ when $p \in S$.

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