
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
Eight teams play a soccer tournament where each team plays every other team exactly once. There are no ties; each team has a $50\%$ chance of winning any game, and the outcomes are independent. Winners earn a point, losers get none. Team $X$ defeats team $Y$ in the first game. Calculate the probability that team $X$ finishes with more points than team $Y$, expressed as a fraction in lowest terms. Find the sum of the numerator and denominator.

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