
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 school wants to implement the "Twenty Great Spirits" and develop students' physical fitness. In order to understand the situation of students' "daily exercise time" (referred to as "exercise time"), a random survey of 50 students' "exercise time" was conducted at this school, and statistics were compiled to draw the following table:

| Group | "Exercise Time" $t$ / minutes | Frequency | Average "Exercise Time" within the group / minutes |
|-------|-------------------------------|-----------|--------------------------------------------------|
| $A$   | $t < 20$                      | $4$       | $15$                                             |
| $B$   | $20 \leq t < 30$              | $8$       | $25$                                             |
| $C$   | $30 \leq t < 40$              | $20$      | $35$                                             |
| $D$   | $t \geq 40$                   | $18$      | $50$                                             |

Based on the above information, answer the following questions:
$(1)$ The median of the "exercise time" of these 50 students falls in the ______ group;
$(2)$ Find the average "exercise time" of these 50 students;
$(3)$ If the school has 3600 students, estimate the number of students in the school who have an "exercise time" of at least 30 minutes.

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