
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 black bag, there are 8 balls, all identical except for their colors. There are 3 white balls, 3 yellow balls, 1 red ball, and 1 green ball. Two players, A and B, are playing a game where they draw balls from the bag to earn points. Each white ball is worth 1 point, each yellow ball is worth 2 points, each red ball is worth 3 points, and each green ball is worth 4 points. The player with the higher score wins. The rules of the game are as follows: 
1. Only one player can draw a ball at a time.
2. The drawn ball is not returned to the bag.
3. The player first draws one ball from the bag. If it is a green ball, they draw 2 more balls. If it is not a green ball, they draw 3 more balls. The player's score is the sum of the points of the balls drawn in the two rounds.
4. The remaining balls go to the other player, who earns points equal to the sum of the remaining balls.

(1) If player A draws a green ball in the first round, find the probability that A's score is not less than B's score.
(2) If player B draws a red ball in the first round, find the probability distribution and the mathematical expectation of B's score, denoted by ξ.

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