
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 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 100 viewers who chose to shop in the live broadcast rooms of these two companies during a certain period of time, and the following data was obtained:  

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

Please complete the table and determine whether there is a 99.9% certainty that the choice of shopping in which live broadcast room is related to the user's age.  
$(2)$ If Xiao Li chooses to shop in one of Company A or Company B's live broadcast rooms for two consecutive days, and on the first day, with equal probability, he chooses to shop in either Company A or Company B's live broadcast room. If he shops in Company A's live broadcast room on the first day, the probability of him shopping in Company A's live broadcast room on the second day is 0.7; if he shops in Company B's live broadcast room on the first day, the probability of him shopping in Company A's live broadcast room on the second day is 0.8. Find the probability that Xiao Li will shop in Company B's live broadcast room on the second day.  
Reference formula: ${\chi^2}=\frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$, where $n=a+b+c+d$. ${\chi^2}$ critical value table:  

| $P(\chi^2 \geq k)$ | 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}
user