
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 country, some cities are connected by direct bus routes. The residents of this country consider an odd natural number \( n \) to be unlucky, so there does not exist a cyclic route consisting of exactly \( n \) trips (and not entering any city twice). Nonetheless, a path containing exactly 100 trips (and not passing through any city twice) exists from any city to any other city. For which odd \( n \) is this possible?

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