
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
We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both 1. At step \( n > 1 \), place a point at the midpoint of each arc created at step \( n-1 \) and label it with the sum of the labels at the two adjacent points. What is the total sum of the labels after step \( n \)?

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