
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 payment for the stay in the form of rings from a gold chain the traveler wore on his wrist. However, the owner set a condition that the payment must be made daily: each day, the owner should have one more ring than the previous day. The closed chain had 11 rings, and the traveler intended to stay exactly 11 days, so he 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 consist of for the traveler to stay at the inn for the maximum number of days under the condition that he can cut only $n$ rings?

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