
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 country A with 2011 cities and country B with 2011 cities, there is a unique bidirectional flight route between any city in country A and any city in country B. There are no routes between cities within the same country, and at most 19 different airlines operate these routes. Find the largest integer \( k \) such that no matter how the routes are assigned, there will always exist \( k \) cities that are connected (where you can travel from one city to another, although not necessarily directly) and all the routes are operated by the same airline.

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