
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
April 23rd of each year is designated as "World Book Day" by UNESCO, also known as "World Book and Copyright Day." School $A$ conducted a random survey of $n$ students to understand their extracurricular reading habits and found that the daily average reading time (in minutes) of these students falls within the range of $\left[0,120\right]$. Based on the daily average reading time of these $n$ students, the sample data is grouped as follows: $\left[0,20\right)$, $\left[20,40\right)$, $\left[40,60\right)$, $\left[60,80\right)$, $\left[80,100\right)$, $\left[100,120\right]$, and the frequency distribution table is shown below:

| Group         | Frequency | Frequency Rate |
|---------------|-----------|----------------|
| $\left[0,20\right)$   | $4$       | $f_{1}$         |
| $\left[20,40\right)$  | $10$      | $0.1$           |
| $\left[40,60\right)$  | $46$      | $f_{2}$         |
| $\left[60,80\right)$  | $a$       | $f_{3}$         |
| $\left[80,100\right)$ | $20$      | $f_{4}$         |
| $\left[100,120\right]$| $4$       | $f_{5}$         |

$(1)$ Find the values of $n$ and $f_{3}$;  
$(2)$ If a stratified random sampling method is used to select $10$ students from those with daily average reading times in the ranges $\left[60,80\right)$, $\left[80,100\right)$, $\left[100,120\right]$, and then one student is randomly selected from the $10$ students for a reading experience sharing, find the probability that the student selected for the reading experience sharing has a daily average reading time of at least $80$ minutes;  
$(3)$ Now, select the $10$ students with the longest daily average reading times from these $n$ students as "Reading Masters." Calculate the minimum daily average reading time required to become a "Reading Master."

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