
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]


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In 2025, our province will implement the "3+1+2" college entrance examination mode, where "3" refers to Chinese, Mathematics, and Foreign Language participating in the national unified examination, and the selective examination subjects are Political Science, History, Geography, Physics, Chemistry, and Biology. Candidates need to choose 2 out of Physics and History first, and then choose 2 out of Political Science, Geography, Chemistry, and Biology to form their own college entrance examination combination.

$(1)$ If a student randomly selects subjects according to the plan, find the probability that the student selects the "History, Political Science, Geography" combination exactly.

$(2)$ Since Physics and History must choose 1 subject, a school wants to understand the subject selection needs of the first-year students. A survey was conducted on 100 first-year students, and the following statistical data was obtained. Write down the values of $a$ and $d$ in the following contingency table, and determine if there is a 95% confidence that "subject selection is related to gender"?

|         | Choose Physics | Choose History | Total |
|---------|-----------------|----------------|-------|
| Boys    | $a$             | $10$           |       |
| Girls   | $30$            | $d$            |       |
| Total   |                 | $30$           |       |

Given: ${K}^{2}=\frac{n{(ad-bc)}^{2}}{(a+b)(c+d)(a+c)(b+d)}$.

| $P(K^{2} \gt k_{0})$ | $0.10$ | $0.05$ | $0.025$ | $0.01$ | $0.005$ |
|----------------------|--------|--------|---------|--------|---------|
| $k_{0}$              | $2.706$| $3.841$| $5.024$ | $6.635$| $7.879$|

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