
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
In a certain country, there are 100 cities. Some pairs of cities are connected by two-way flights, with no more than $n$ flights per city. A ticket can only be purchased for a route connecting two different cities consisting of two flights with a layover in a third city (routes differing in travel direction or layover city are considered different). It turned out that tickets are sold for a total of 1000 different routes. What is the minimum value of $n$ for which this is possible?

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