
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
Nine teams play a soccer tournament where each team plays every other team exactly once. No ties occur, each match results in a win or a loss with each team having a 50% chance of winning any given match, and the outcomes are independent. The winner of each match earns a point while the loser gets none. At the conclusion of the tournament, team $C$ has played and beaten teams $D, E,$ and $F$ but lost to team $G$. The probability that team $C$ finishes with more points than teams $D, E,$ and $F$ but fewer than $G$ is $x/y,$ where $x$ and $y$ are coprime. Find $x+y$.

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