
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
Six students A, B, C, D, E, and F hold a chess competition, adopting a round-robin system, meaning each pair of participants plays exactly one game against each other. On the first day, A and B each participated in 3 games, C and D each participated in 4 games, E participated in 2 games, and A did not play against C, B did not play against D. Determine the total number of games F participated in on the first day.

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