
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, each player played exactly one game against each of the other players. In each game, the winner was awarded 1 point, the loser got 0 points, and each player earned \( \frac{1}{2} \) point if the game was a tie. After the tournament, it was found that exactly half of the points earned by each player were earned against the five players with the least number of points. (In particular, each of the five lowest scoring players earned half of their points against the other four of the five). What was the total number of players in the tournament?

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