
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
Five airlines operate in a country consisting of  $36$  cities. Between any pair of cities exactly one airline operates two way 
flights. If some airlines operates between cities  $A,B$  and  $B,C$  we say that the ordered triple  $A,B,C$  is properly-connected. Determine the largest possible value of  $k$  such that no matter how these flights are arranged there are at least  $k$  properly-connected triples.

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