
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
Michelle is drawing segments in the plane. She begins from the origin facing up the  $y$ -axis and draws a segment of length  $1$ . Now, she rotates her direction by  $120^\circ$ , with equal probability clockwise or counterclockwise, and draws another segment of length  $1$  beginning from the end of the previous segment. She then continues this until she hits an already drawn segment. What is the expected number of segments she has drawn when this happens?

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