
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 commercial football tournament, five teams participated. Each team was supposed to play exactly one match with each of the other teams. Due to financial difficulties, the organizers canceled some games. As a result, all teams ended up with a different number of points, and none of the teams had zero points in the points column. What is the minimum number of games that could have been played in the tournament if a win awarded three points, a draw one point, and a loss zero points?

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