We note that $$997\equiv -3\pmod{1000},$$and $$(-3)\cdot 333 = -999 = -1000+1\equiv 1\pmod{1000}.$$Therefore, $$997\cdot 333\equiv 1\pmod{1000},$$and the inverse of $997$ modulo $1000$ is $\boxed{333}$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Compute $997^{-1}$ modulo $1000$. Express your answer as an integer from $0$ to $999$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
333
Compute $997^{-1}$ modulo $1000$. Express your answer as an integer from $0$ to $999$.
test