
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 high school offers four school-based elective courses for the second year of high school: "Mathematical Modeling," "Film Appreciation," "Classic Reading," and "English Writing." Students A, B, and C plan to randomly select one of the four courses for study. It is known that the choices of the three students do not affect each other, and each student has an equal probability of choosing each course.  
$(1)$ Find the probability that the three students choose different courses.  
$(2)$ If at least two students choose "Mathematical Modeling," how many different course selection options are there for the three students?

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