A four-digit number $n$ is said to be "literally 1434" if, when every digit is replaced by its remainder when divided by $5$, the result is $1434$. For example, $1984$ is "literally 1434" because $1 \mod 5$ is $1$, $9 \mod 5$ is $4$, $8 \mod 5$ is $3$, and $4 \mod 5$ is $4$. Find the sum of all four-digit positive integers that are "literally 1434".

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