
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
Consider a hexagon with vertices labeled  $M$ ,  $M$ ,  $A$ ,  $T$ ,  $H$ ,  $S$  in that order. Clayton starts at the  $M$  adjacent to  $M$  and  $A$ , and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability  $\frac{1}{2}$ , and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters  $M, A, T$ , and  $H$  in that order, not necessarily consecutively (for example, one valid string might be  $MAMMSHTH$ .) What is the expected length of the string Clayton wrote?

*Proposed by Andrew Milas and Andrew Wu*

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