
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
Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\frac{m}{n}$ where $m$ and $n$ are coprime integers and find $\log_2 n$.

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