
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-phone management," while those who spent less than 75 hours were considered to have "good self-phone management."

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

|                   | Good Self-Management | Poor Self-Management | Total |
|-------------------|----------------------|----------------------|-------|
| Male Students     |                      |                      |       |
| Female Students   |                      | $12$                 | $40$  |
| Total             |                      |                      |       |

$(2)$ Based on the conditions in $(1)$, stratified sampling was used to select 5 individuals from those with "poor self-phone management," with 1 male and 1 female among them who enjoy sports. Now, randomly select 3 individuals from these 5 with "poor self-phone management." Find the probability that all 3 individuals are of both genders and at least one of them enjoys sports.

Critical values for the $K^2$ independence test:

| $P(K^2 \geq k_0)$ | $0.10$ | $0.05$ | $0.010$ | $0.001$ |
|-------------------|--------|--------|---------|---------|
| $k_0$             | $2.706$| $3.841$| $6.635$ | $10.828$|

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