There are $63$ houses at distances of $1, 2, 3, \ldots, 63$ km from the North Pole, respectively. Santa Claus wants to distribute vaccines to each house. To do so, he will have his $63$ assistants, elves named $E_1, E_2, \ldots, E_{63}$, deliver the vaccine to each house; each elf will deliver the vaccine to exactly one house and never return. Suppose that elf $E_n$ takes $n$ minutes to travel $1$ km for each $n = 1, 2, \ldots, 63$, and that all elves leave the North Pole simultaneously. What is the minimum amount of time to complete the delivery?

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