
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 high school has created three clubs, "Calligraphy," "Poetry," and "Science," based on students' interests. According to statistics, the probabilities of a new student passing the selection exams and entering these three clubs are $m$, $\frac{1}{3}$, and $n$ respectively. It is known that the probability of entering all three clubs is $\frac{1}{24}$, the probability of entering at least one club is $\frac{3}{4}$, and $m > n$.  
$(1)$ Find the values of $m$ and $n$;  
$(2)$ Based on the activities of the three clubs, the school awards 1 credit for entering the "Calligraphy" club, 2 credits for entering the "Poetry" club, and 3 credits for entering the "Science" club. Find the probability that the new student will earn at least 4 credits in terms of club activities.

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