
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
At a mathematical competition  $n$  students work on  $6$  problems each one with three possible answers. After the competition, the Jury found that for every two students the number of the problems, for which these students have the same answers, is  $0$  or  $2$ .  Find the maximum possible value of  $n$ .

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