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.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
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.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
0
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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