
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
Two players are engaged in a table tennis match, which is a best of three format. In each game, the player who first reaches 11 points and has at least 2 points more than the opponent wins. If both players reach 10 points, the game continues until one player leads by 2 points. The sum of the scores of the two players across all games is 31 points. One player won the first game and ultimately won the match. How many possible scores are there for the second game?

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