<|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
 The sequence \( 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots \) is formed as follows: write down infinitely many '1's, insert a '2' between the first and the second '1's, insert two '2's between the second and the third '1's, insert three '2's between the third and the fourth '1's, and so on. If \( a_{n} \) denotes the \( n \)-th term of the sequence, find the value of \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2013} a_{2014} \). 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
7806
The sequence \( 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots \) is formed as follows: write down infinitely many '1's, insert a '2' between the first and the second '1's, insert two '2's between the second and the third '1's, insert three '2's between the third and the fourth '1's, and so on. If \( a_{n} \) denotes the \( n \)-th term of the sequence, find the value of \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2013} a_{2014} \).
test