
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 chess tournament, all players faced each other twice and scored 1 point for a win, half a point for a draw, and zero points for a loss. The winner was the one who scored the most points. Luíza, a curious mathematician, found a paper stating that the sum of the scores of all participants was 210 points.

a) How many players participated in the tournament?

b) Luíza noticed that the tournament logo depicted a knight in the center, with black squares illustrating the knight's possible moves. She wondered:

- If the board were infinite and the knight continued its pattern of movement, onto how many different squares could the knight move in two moves?

Initially, she thought “64” but quickly realized she was wrong. Now, it's your turn to correctly answer the question Luíza thought of.

c) Luíza received a message from a friend who participated in the tournament, informing her that he scored 12 points. Luíza replied: “- Don’t be sad, you will win the next championship!” How did Luíza know that her friend did not win the tournament?

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