<|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
 Let the sequence \( a_1, a_2, \cdots, a_n, \cdots \) be defined such that \( a_1 = a_2 = 1, a_3 = 2 \), and for any positive integer \( n \), \( a_n \cdot a_{n+1} \cdot a_{n+2} \neq 1 \). Also, given that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \), find the value of \( a_1 + a_2 + \cdots + a_{100} \). 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
200
Let the sequence \( a_1, a_2, \cdots, a_n, \cdots \) be defined such that \( a_1 = a_2 = 1, a_3 = 2 \), and for any positive integer \( n \), \( a_n \cdot a_{n+1} \cdot a_{n+2} \neq 1 \). Also, given that \( a_n \cdot a_{n+1} \cdot a_{n+2} \cdot a_{n+3} = a_n + a_{n+1} + a_{n+2} + a_{n+3} \), find the value of \( a_1 + a_2 + \cdots + a_{100} \).
test