
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, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).

a) What is the sum of the scores of the players in \( S \)?

b) Determine how many participants are in the tournament.

Note: Each player plays only once with each opponent.

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