
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 in a league where each team plays every other team exactly once, and every game results in a win or loss with no ties. Each game is independent with a $50\%$ chance of either team winning. Determine the probability that no two teams end up with the same number of total victories, expressed as $\frac{p}{q}$ where $p$ and $q$ are coprime positive integers. Find $\log_2 q.$

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