What is the unique three-digit positive integer $x$ satisfying $$100x\equiv 1\pmod{997}~?$$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
What is the unique three-digit positive integer $x$ satisfying $$100x\equiv 1\pmod{997}~?$$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
668
What is the unique three-digit positive integer $x$ satisfying $$100x\equiv 1\pmod{997}~?$$
train