
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
A set of $n$ people participate in an online rugby tournament. Each person may be a member of any number of $6$-player teams, but no two teams may have exactly the same $6$ members. The site statistics show an intriguing fact: The average, over all subsets of size $10$ of the set of $n$ participants, of the number of complete teams whose members are among those $10$ people is equal to the reciprocal of the average, over all subsets of size $9$ of the set of $n$ participants, of the number of complete teams whose members are among those $9$ people. How many values of $n$, $10 \leq n \leq 2017$, can be the number of participants?

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