
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
At a market, a buyer and a seller each have four exotic coins. You are allowed to label each of the eight coins with any positive integer value in cents. The labeling is called \(n\)-efficient if for any integer \(k, 1 \leq k \leq n\), it is possible for the buyer and the seller to give each other some of their coins in such a way that, as a net result, the buyer has paid the seller \(k\) cents. Find the greatest positive integer \(n\) such that an \(n\)-efficient labeling of the coins exists.

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