
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 summer math research study group at a certain experimental high school conducted a survey on the amount of time (in hours) students spent on their phones in August. They randomly surveyed 100 classmates and organized the data into the following table:

| Time Spent on Phone | $[0,15)$ | $[15,30)$ | $[30,45)$ | $[45,60)$ | $[60,75)$ | $[75,90)$ | $[90,+\infty)$ |
|---------------------|----------|-----------|-----------|-----------|-----------|-----------|-----------------|
| Number of Students  | $1$      | $12$      | $28$      | $24$      | $15$      | $13$      | $7$             |

Students who spent 75 hours or more on their phones in August were considered to have "poor self-management of phone usage," while those who spent less than 75 hours were considered to have "good self-management of phone usage."

$(1)$ Please complete the $2\times 2$ contingency table below based on the given conditions and determine if there is a $99\%$ confidence that "self-management of phone usage is related to gender":

|                   | Good Self-Management | Poor Self-Management | Total |
|-------------------|----------------------|----------------------|-------|
| Male Students     | $52$                 | $8$                  | $60$  |
| Female Students   | $28$                 | $12$                 | $40$  |
| Total             | $80$                 | $20$                 | $100$ |

$(2)$ The school's physical education teacher selected 2 male students and 1 female student from those with poor self-management of phone usage for basketball training. It is known that the probability of a male student scoring a basket is $\frac{2}{3}$ and the probability of a female student scoring a basket is $\frac{1}{2}$. Each person takes one shot, assuming the shots are independent. Find the distribution table and mathematical expectation of the total number of successful shots made by the 3 students.

Appendix: ${\chi^2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$, where $n=a+b+c+d$. 

$\chi^2$ independence test critical value table:

| $P(\chi^2\geqslant \chi_0)$ | $0.10$ | $0.05$ | $0.010$ | $0.001$ |
|-----------------------------|--------|--------|---------|---------|
| $\chi_0$                    | $2.706$| $3.841$| $6.635$ | $10.828$|

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