
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 the Women's Singles Badminton final of the 2008 Beijing Olympics, Chinese athlete Zhang Ning won 2:1 against her teammate Xie Xingfang, who was ranked number one in the world, to retain her Olympic championship. The badminton match follows a "best-of-three" format (i.e., the player who first wins two sets wins the match, and the match ends), with each set independent of the others. According to the analysis of their previous head-to-head records, Xie Xingfang had a winning probability of 0.6 in each of the first two sets. However, if the score reached 1:1 after two sets, Zhang Ning, with her strong mental strength, had a winning probability of 0.6 in the third set. If Zhang Ning and Xie Xingfang were to meet again in a match,
1. Calculate the probability that Zhang Ning wins 2:1.
2. Let ξ represent the net winning sets for Zhang Ning. Calculate the probability distribution of ξ and Eξ (the expected value of ξ).

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