What is the residue modulo $13$ of the sum of the modulo $13$ inverses of the first $12$ positive integers?

Express your answer as an integer from $0$ to $12$, inclusive.
Question:
What is the residue modulo $13$ of the sum of the modulo $13$ inverses of the first $12$ positive integers?

Express your answer as an integer from $0$ to $12$, inclusive.
Answer:
Let's think step by step.

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What is the residue modulo $13$ of the sum of the modulo $13$ inverses of the first $12$ positive integers?

Express your answer as an integer from $0$ to $12$, inclusive.
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