
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 people, A and B, are playing a game. Each time one player wins, they get 1 point, and the loser gets 0 points. The game ends as soon as one player's score exceeds the other player's score by 2 points. The maximum number of rounds allowed is 20; if the game reaches 20 rounds, the player with the higher score wins, and if the scores are equal, the game is considered a draw. The probability of A winning a single round is \( p \) (where \( 0 < p < 1 \)), and the probability of B winning a single round is \( q = 1 - p \). Assuming the results of each round are independent, the game ends after \( \xi \) rounds. Determine the range of the expected value \( E\xi \).

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