
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 the fictional country of Mahishmati, there are  $50$  cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city  $A$ , an ordered list of cities  $C_1,\ldots, C_{50}$  is called an *antitour* from  $A$  if


- every city (including  $A$ ) appears in the list exactly once, and
- for each  $k\in \{1,2,\ldots, 50\}$ , it is impossible to go from  $A$  to  $C_k$  by a sequence of exactly  $k$  (not necessarily distinct) flights.

Baahubali notices that there is an antitour from  $A$  for any city  $A$ . Further, he can take a sequence of flights, starting from the capital and passing through each city exactly once. Find the least possible total number of antitours from the capital city.

*Proposed by Sutanay Bhattacharya*

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