
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
There are 20 villages distributed along the shore of a circular lake, and each village has 20 warriors. In a martial arts competition, every two warriors from different villages will compete once. If in the competitions between all the warriors of villages A and B, there are at least $\mathrm{k}$ matches where the warriors from village A win, then village A is considered stronger than village B. It is known that each village is stronger than the village adjacent to it in a clockwise direction. Find the maximum possible value of $\mathrm{k}$ (assuming all warriors have different skill levels and the stronger warrior always wins in each match).

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