
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 round-robin tournament has six competititors. Each round between two players is equally likely to result in a win for a given player, a loss for that player, or a tie. The results of the tournament are \textit{nice} if for all triples of distinct players  $(A, B, C)$ ,
  
  1. If  $A$  beat  $B$  and  $B$  beat  $C$ , then  $A$  also beat  $C$ ;
  
  2. If  $A$  and  $B$  tied, then either  $C$  beat both  $A$  and  $B$ , or  $C$  lost to both  $A$  and  $B$ .
  
  The probability that the results of the tournament are  $\textit{nice}$  is  $p = \tfrac{m}{n}$ , for coprime positive integers  $m$  and  $n$ . Find  $m$ .

*Proposed by Michael Tang*

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