
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
20 different villages are located along the coast of a circular island. Each of these villages has 20 fighters, with all 400 fighters having different strengths.

Two neighboring villages $A$ and $B$ now have a competition in which each of the 20 fighters from village $A$ competes with each of the 20 fighters from village $B$. The stronger fighter wins. We say that village $A$ is stronger than village $B$ if a fighter from village $A$ wins at least $k$ of the 400 fights.

It turns out that each village is stronger than its neighboring village in a clockwise direction. Determine the maximum value of $k$ so that this can be the case.

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