Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions:  
1. $f(mn) = f(m)f(n)$  
2. $f(m) < f(n)$ if $m < n$  
3. $f(2) = 2$  

What is the sum of $\sum_{k=1}^{20}f(k)$?

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