
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 a football league, there are  $n\geq 6$  teams. Each team has a homecourt jersey and a road jersey with different color. When two teams play, the home team always wear homecourt jersey and the road team wear their homecourt jersey if the color is different from the home team's homecourt jersey, or otherwise the road team shall wear their road jersey. It is required that in any two games with 4 different teams, the 4 teams' jerseys have at least 3 different color. Find the least number of color that the  $n$  teams'  $2n$  jerseys may use.

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