<|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 that $\{a_n\}$ is an arithmetic sequence, and $a_1+a_3=8$, $a_2+a_4=12$. Let $S_n$ denote the sum of the first $n$ terms of $\{a_n\}$. If $a_1$, $a_k$, $S_{k+2}$ form a geometric sequence, then the value of the positive integer $k$ is \_\_\_\_\_\_. 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
6
Given that $\{a_n\}$ is an arithmetic sequence, and $a_1+a_3=8$, $a_2+a_4=12$. Let $S_n$ denote the sum of the first $n$ terms of $\{a_n\}$. If $a_1$, $a_k$, $S_{k+2}$ form a geometric sequence, then the value of the positive integer $k$ is \_\_\_\_\_\_.
test