
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
On the shore of a round island Somewhere are located 20 villages, each inhabited by 20 wrestlers. A tournament was held in which each wrestler competed against every wrestler from all other villages. Village A is considered stronger than village B if at least $k$ matches between wrestlers from these villages end with a victory for a wrestler from village A. It turned out that each village is stronger than the next village clockwise. What is the maximum possible value of $k$? (All wrestlers have different strengths, and the strongest always wins in a match.)

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