
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
The Harbin Ice and Snow World in China is a large-scale ice and snow art project launched by the Harbin Municipal Government, showcasing the ice and snow culture and charm of the northern city of Harbin. The annual activities include ice harvesting and ice carving. There are currently three work teams, Team A, Team B, and Team C, responsible for the above activities. During the carving process, some ice blocks will be damaged, and if damaged, they cannot be used, while undamaged blocks are all used. It is known that Teams A, B, and C account for 30%, 30%, and 40% of the total ice harvested, respectively. The utilization rates of the ice harvested by Teams A, B, and C are 0.8, 0.75, and 0.6, respectively. 

$(1)$ From the harvested ice blocks, three blocks are randomly selected with replacement. Let $\xi$ be the number of times the ice blocks harvested by Team C are selected. Find the probability distribution and expectation of the random variable $\xi$.

$(2)$ Given that the harvested ice blocks can be used after carving, find the probability that a block was harvested by Team B.

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