
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
The Tiangong Space Station is a national space laboratory built by our country, consisting of the Tianhe core module, the Wentian experimental module, and the Mengtian experimental module. It has entered a long-term manned mode. Combining relevant knowledge of the space station, a teacher from a vocational school designed a programming training based on the space station theme. The training content consists of 10 independent programming questions such as "Space Launch," "Custom Roaming," "Full-size Solar Energy," and "Space Transport." The training requires each student to choose two different questions for programming practice, and the choices of students do not affect each other. The teacher divided the class students into four groups, designating them as Group A, Group B, Group C, and Group D.

$(1)$ Find the probability that at least one of Group A, Group B, Group C, or Group D chooses "Space Launch";
$(2)$ Let $X$ be the number of people in these four groups who choose "Space Launch." Find the probability distribution and mathematical expectation of $X$;
$(3)$ If there are $n$ students participating in the programming training in the class (where $n$ is a positive integer divisible by $5$), then how many people in these $n$ students are most likely to choose "Space Launch"?

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