
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 total of 1000 students are enrolled in College A of a university, with 640 male students and 360 female students. In order to understand the students' participation in running activities, the sports club of the college conducted a stratified sampling by gender, selecting a certain number of students as samples from all students of the college. The club then conducted a statistical analysis of the cumulative running mileage of each student in May, as shown in the table below.

| Running Mileage $s$ (km) | $0 \leq s < 30$ | $30 \leq s < 60$ | $60 \leq s < 90$ | $s \geq 90$ |
|--------------------------|-----------------|------------------|------------------|-------------|
| Male Students (Number)   | $a$             | $12$             | $10$             | $5$         |
| Female Students (Number) | $6$             | $6$              | $4$              | $2$         |

$(1)$ Find the value of $a$ and estimate the number of male students in College A with a cumulative running mileage in the range $[0, 30)$ km in May.  
$(2)$ From the sample of College A, three students with a cumulative running mileage of at least $90$ km in May are randomly selected. Let the number of male students among them be denoted as $X$. Find the distribution and expectation of $X$.

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