
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 randomly selected several students to investigate the daily physical exercise time of students in the school. They obtained data on the daily physical exercise time (unit: minutes) and organized and described the data. Some information is as follows:
- $a$. Distribution of daily physical exercise time:
| Daily Exercise Time $x$ (minutes) | Frequency (people) | Percentage |
|-----------------------------------|--------------------|------------|
| $60\leqslant x \lt 70$            | $14$               | $14\%$     |
| $70\leqslant x \lt 80$            | $40$               | $m$        |
| $80\leqslant x \lt 90$            | $35$               | $35\%$     |
| $x\geqslant 90$                   | $n$                | $11\%$     |

- $b$. The daily physical exercise time in the group $80\leqslant x \lt 90$ is: $80$, $81$, $81$, $81$, $82$, $82$, $83$, $83$, $84$, $84$, $84$, $84$, $84$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $85$, $86$, $87$, $87$, $87$, $87$, $87$, $88$, $88$, $88$, $89$, $89$, $89$, $89$, $89$.

Based on the above information, answer the following questions:
$(1)$ In the table, $m=$______, $n=$______.
$(2)$ If the school has a total of $1000$ students, estimate the number of students in the school who exercise for at least $80$ minutes per day.
$(3)$ The school is planning to set a time standard $p$ (unit: minutes) to commend students who exercise for at least $p$ minutes per day. If $25\%$ of the students are to be commended, what value can $p$ be?

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