<|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
 
	 Starting with an empty string, we create a string by repeatedly appending one of the letters  $H$ ,  $M$ ,  $T$  with probabilities  $\frac 14$ ,  $\frac 12$ ,  $\frac 14$ , respectively, until the letter  $M$  appears twice consecutively. What is the expected value of the length of the resulting string? 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

	 Starting with an empty string, we create a string by repeatedly appending one of the letters  $H$ ,  $M$ ,  $T$  with probabilities  $\frac 14$ ,  $\frac 12$ ,  $\frac 14$ , respectively, until the letter  $M$  appears twice consecutively. What is the expected value of the length of the resulting string?
test