
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, A and B, are involved in a game where the winner of each round scores 1 point and the loser scores 0 points. The game ends when one player's score exceeds the other's score by 2 points. Additionally, the game can have a maximum of 20 rounds. If the game reaches 20 rounds without either player achieving the 2-point lead, the player with the higher score wins; in the case of a tie, the game ends in a draw. The probability of player A winning each round is \( p \) (\( 0<p<1 \)), and the probability of player B winning each round is \( q = 1-p \). Assuming the results of each round are independent, and the game ends after \( \xi \) rounds, find the expected value \( E[\xi] \) and determine its range of values.

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