
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
Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the ﬁrst game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the ﬁnal hole. Garfunkle went on to score  $8$  total victories, but no ties at all. Meanwhile, Simon wound up with exactly  $8$  points, including the point for a victory over Garfunkle. Amazingly, every other player at the tournament scored exactly  $n$ . Find the sum of all possible values of  $n$ .

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