
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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Given that a certain high school has 450 students (250 boys and 200 girls). In order to provide students with a richer campus cultural life, the school has added two new school-based courses, A and B. Students can choose one of these two courses based on their interests. The school has collected data on students' course selections and created the following 2x2 contingency table.

|          | Choose Course A | Choose Course B | Total |
|----------|-----------------|-----------------|-------|
| Boys     |                 | 150             |       |
| Girls    | 50              |                 |       |
| Total    |                 |                 |       |

$(1)$ Please complete the contingency table and determine whether there is a 99.9% certainty that the choice of course is related to gender. Explain your reasoning.  
$(2)$ From all the boys, conduct a stratified random sampling based on the course selection data in the contingency table. Select 10 boys, and then select 3 boys from these 10 for a questionnaire survey. Let X be the number of boys among these 3 who chose Course A. Find the distribution table and the expected value of X.

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

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