
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
During the Spring Festival in 2023, in order to further leverage the positive role of e-commerce in the active consumer market and ensure that the people spend a safe, healthy, happy, and peaceful Chinese New Year, Company A and Company B simultaneously launched discount promotions and live streaming activities with New Year goods on a certain shopping platform. Company A and Company B sell similar products and compete with each other.  
$(1)$ Now, a survey was conducted on the shopping choices of 100 viewers who watched the live streams of these two companies during a certain period of time, and the following data was obtained:  

|                   | Choose Company A | Choose Company B | Total |
|-------------------|------------------|------------------|-------|
| Age 19-24 years   | 40               | 10               | 50    |
| Age 25-34 years   | 20               | 30               | 50    |
| Total             | 60               | 40               | 100   |

Can we be $99.9\%$ certain that the choice of shopping in which live stream room is related to the user's age?  
$(2)$ If Xiao Li chooses to shop in one of the live stream rooms of Company A or Company B for two consecutive days, and on the first day, he equally likely chooses one of the two live stream rooms for shopping, with a probability of $0.7$ to shop in Company A's live stream room on the second day if he shopped there on the first day, and a probability of $0.8$ to shop in Company A's live stream room on the second day if he shopped in Company B's live stream room on the first day, what is the probability that Xiao Li will shop in Company B's live stream room on the second day?  
$(3)$ During New Year's Day, Company A's shopping platform live stream room conducted a "flash sale" activity. Assuming that the probability of each person successfully placing an order in the live stream room is $p (0<p<1)$, and the success or failure of each person's order is independent. If five people are randomly selected from the live stream room, and the probability of exactly $2$ people successfully placing an order among the five people is denoted as $f(p)$, find the maximum point $p_{0}$ of $f(p)$.  
Reference formula: $K^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.  
Common small probability values and corresponding critical values in $K^{2}$ independence test table:  

| $P(K^{2}\geqslant k)$ | $0.10$ | $0.05$ | $0.01$ | $0.005$ | $0.001$ |
|-----------------------|--------|--------|--------|---------|---------|
| $k$                   | $2.706$| $3.841$| $6.635$| $7.879$ | $10.828$|

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