
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

...

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system
The newly revised "Sports Law of the People's Republic of China" will be implemented from January 1, 2023. It is of great significance for leading the high-quality development of China's sports industry, promoting the construction of a sports power and a healthy China. A certain middle school in Ganzhou conducted a survey on the relationship between students' gender and whether they like playing badminton. Using simple random sampling, 200 students were selected as samples. The probability of female students in the sample is $\frac{1}{2}$. Let $A=$ "like badminton," "student is female." According to statistics, $P(A|\overline{B})=\frac{2}{5}$, $P(B|\overline{A})=\frac{3}{5}$.
$(1)$ Please complete the following $2\times 2$ contingency table based on the known conditions:
|              | Like Badminton | Dislike Badminton | Total |
|--------------|----------------|-------------------|-------|
| Male         |                |                   |       |
| Female       |                |                   |       |
| Total        |                |                   |       |

$(2)$ Based on this, how confident can we be in determining the relationship between students' gender and whether they like playing badminton?
Given: $K^2=\frac{n(ad-bc)^2}{(a+b)(c+d)(a+c)(b+d)}$, where $n=a+b+c+d$.
Reference data:
| $P(K^{2}\geqslant k_{0})$ | $0.100$ | $0.050$ | $0.010$ | $0.001$ |
|---------------------------|---------|---------|---------|---------|
| $k_{0}$                   | $2.706$| $3.841$| $6.635$| $10.828$|

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