
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
A well-known cake shop in a certain place has drawn up the following frequency distribution table based on the sales records of a certain type of cake in $100$ days:<br/>

| Daily Sales (unit: pieces) | $\left[0,50\right)$ | $\left[50,100\right)$ | $\left[100,150\right)$ | $\left[150,200\right)$ | $\left[200,250\right)$ |
|---------------------------|---------------------|-----------------------|------------------------|------------------------|------------------------|
| Frequency                | $15$                | $25$                  | $30$                   | $20$                   | $10$                   |

Treating the probability of daily sales falling into each group as the probability, and assuming that the daily sales volume is independent of each other.<br/>
$(1)$ Find the probability that in the next consecutive $3$ days, there are at least $2$ days with daily sales of at least $100$ pieces each and the daily sales of the other day are less than $50$ pieces;<br/>
$(2)$ Let $\xi$ represent the number of days in the next $3$ days with daily sales of at least $150$ pieces, find the probability distribution of the random variable $\xi$, the mean $E(\xi)$, and the variance $D(\xi)$.

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