<|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
 Given the set \( T = \{1, 2, \cdots, 2010\} \), for each non-empty subset of \( T \), compute the reciprocal of the product of its elements. Find the sum of all these reciprocals. 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
2010
Given the set \( T = \{1, 2, \cdots, 2010\} \), for each non-empty subset of \( T \), compute the reciprocal of the product of its elements. Find the sum of all these reciprocals.
test