
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
Let \( n \) be a positive integer and there are \( 2^n \) weights, each with a different mass. Each time you weigh, you divide the existing weights into two groups of equal numbers and use a balance scale to compare them, then keep the heavier group (if the groups have the same mass, you can keep either group). After \( n \) weighings, one weight remains. What is the minimum rank of this remaining weight in the initial list of \( 2^n \) weights ordered by mass from heaviest to lightest?

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