
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 certain school randomly selected 70 students to investigate whether the elective course preference of high school students is related to gender. The contingency table obtained is as follows:

|         | Preference "Coordinate System and Parametric Equations" | Preference "Inequality Lecture" |
|---------|---------------------------------------------------------|---------------------------------|
| Boys    | 15                                                      | 25                              |
| Girls   | 20                                                      | 10                              |

$(1)$ Based on the data provided in the table, determine whether there is more than a $95\%$ certainty that the preference for "Coordinate System and Parametric Equations" or "Inequality Lecture" is related to gender.<br/>$(2)$ Among the students who prefer "Coordinate System and Parametric Equations," 7 students are selected using stratified sampling by gender. From these 7 students, 3 are selected to participate in a questionnaire survey. Let $\xi$ be the number of boys among these 3 students. Find the probability distribution and the expected value of $\xi$.<br/>Given: ${\chi}^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)},n=a+b+c+d$.<br/>

| $P(\chi^{2} \leq k_{0})$ | 0.100 | 0.050 | 0.010 | 0.005 | 0.001 |
|--------------------------|-------|-------|-------|-------|-------|
| $k_{0}$                  | 2.706 | 3.841 | 6.635 | 7.879 | 10.828 |

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