
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

a) A traveler stopped at an inn, and the owner agreed to accept segments of a gold chain worn by the traveler as payment for lodging. However, the owner stipulated that the payment should be made daily: each day, the owner should receive one more ring than the previous day. The chain, which was closed into a loop, contained 11 rings, and the traveler planned to stay for exactly 11 days. Therefore, the traveler agreed. What is the minimum number of rings he must cut to be able to pay the owner?

b) How many rings must the chain contain to allow the traveler to stay at the inn for the maximum number of days, given that he can cut only $n$ rings?

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