
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 chess tournament there are  $n>2$  players. Every two players play against each other exactly once. It is known that exactly  $n$  games end as a tie. For any set  $S$  of players, including  $A$  and  $B$ , we say that  $A$  *admires*  $B$  *in that set*if
i)  $A$  does not beat  $B$ ; or
ii) there exists a sequence of players  $C_1,C_2,\ldots,C_k$  in  $S$ , such that  $A$  does not beat  $C_1$ ,  $C_k$  does not beat  $B$ , and  $C_i$  does not beat  $C_{i+1}$  for  $1\le i\le k-1$ .
A set of four players is said to be *harmonic* if each of the four players admires everyone else in the set. Find, in terms of  $n$ , the largest possible number of harmonic sets.

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