<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 An electric cable of length 21 meters is cut into 21 pieces. For any two pieces, their lengths differ by no more than a factor of three. What is the smallest value of \(m\) for which there must be two pieces whose lengths differ by no more than \(m\) times? Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
1
An electric cable of length 21 meters is cut into 21 pieces. For any two pieces, their lengths differ by no more than a factor of three. What is the smallest value of \(m\) for which there must be two pieces whose lengths differ by no more than \(m\) times?
test