
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

	 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?

Present the answer in LaTex format: \boxed{Your answer}
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