
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 group of $n$ players participates in a chess tournament. Each player can be part of any number of $4$-player teams, but no two teams can have exactly the same $4$ members. It is noted that the average, over all subsets of size $7$ of the $n$ participants, of the number of complete teams among those $7$ people equals the reciprocal of the average, over all subsets of size $6$ of the $n$ participants, of the number of complete teams among those $6$ people. Determine how many values of $n$, with $7 \leq n \leq 1000$, satisfy this condition.

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