<|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 sequence \(a_1, a_2, \cdots, a_n, \cdots\) satisfying \(a_1 = a_2 = 1\), \(a_3 = 2\), and for any positive integer \(n\), \(a_n a_{n+1} a_{n+2} \neq 1\), and \(a_n a_{n+1} a_{n+2} a_{n+3} = a_n + a_{n-1} + a_{n+2} + a_{n+3}\). 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
Given the sequence \(a_1, a_2, \cdots, a_n, \cdots\) satisfying \(a_1 = a_2 = 1\), \(a_3 = 2\), and for any positive integer \(n\), \(a_n a_{n+1} a_{n+2} \neq 1\), and \(a_n a_{n+1} a_{n+2} a_{n+3} = a_n + a_{n-1} + a_{n+2} + a_{n+3}\). Find the value of \(a_1 + a_2 + \cdots + a_{100}\).
test