
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
To ensure the safety and health of students' diet, the school has upgraded the hardware and dishes in the cafeteria. The upgraded cafeteria dishes have been welcomed by many students, so dining in the school cafeteria has become the dining choice for many students. Students who dine in the cafeteria more than $3$ times a week are considered to "like dining in the cafeteria," while those who dine in the cafeteria $3$ times or less are considered to "not like dining in the cafeteria." In order to understand the students' dining situation in the cafeteria, the school randomly selected $100$ students on campus and the statistical data is as follows:

| Gender | Dining Preference | Total |
|--------|-------------------|-------|
| Male   | Like Cafeteria     | Dislike Cafeteria |  Total |
|--------|-------------------|-------|
| Male   | $40$              | $10$              | $50$  |
| Female | $20$              | $30$              | $50$  |
| Total  | $60$              | $40$              | $100$ |

$(1)$ Based on the significance level $\alpha =0.005$ of the independence test, analyze whether students' preference for dining in the cafeteria is related to gender.  
$(2)$ The student Xiao Lin in the school dines in the school cafeteria on Wednesdays and Fridays. On Wednesdays, he randomly chooses one of the two set meals, Meal $1$ and Meal $2$. If he chooses Meal $1$ on Wednesday, the probability of choosing Meal $1$ on Friday is $0.8$; if he chooses Meal $2$ on Wednesday, the probability of choosing Meal $1$ on Friday is $\frac{1}{3}$. Find the probability that Xiao Lin chooses Meal $2$ on Friday.  
$(3)$ Estimate the probability using frequency. Randomly select $10$ students from the school, let $\xi$ be the number of students who "like dining in the cafeteria" among them, and the probability of the event "$\xi =k$" be $P(\xi =k)$. Find the value of $k$ that maximizes $P(\xi =k)$.  
Reference formula: ${χ^2}=\frac{{n{{({ad-bc})}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$, where $n=a+b+c+d$.  

| $\alpha$ | $0.1$ | $0.05$ | $0.01$ | $0.005$ | $0.001$ |
|----------|-------|--------|--------|---------|---------|
| $x_{\alpha}$ | $2.706$ | $3.841$ | $6.635$ | $7.879$ | $10.828$ |

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