
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 tournament with 10 teams, each team faces each other exactly once. Additionally, there are no ties, and each team has a 50% chance of winning any match. What is the probability that, after counting the scores of the $\frac{10 \cdot 9}{2} = 45$ games, no two players have the same number of wins?

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