
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-five teams play a tournament where each team plays against every other team exactly once, and there are no ties allowed. Each game has a $50\%$ chance of being won by either team. Find the probability that no two teams will end up with the same number of victories in a simplified form $\frac{m}{n}$ where $m$ and $n$ are coprime, and compute $\log_2 n$.

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