
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]


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In the "Charity Day Donation" event, the Youth League Committee of a school in Cuiping District randomly selected the donation amounts of $10$ students for statistical analysis and presented the data in the following table.

| Amount (in yuan) | $5$ | $10$ | $15$ | $20$ |
|------------------|-----|------|------|------|
| Number of Students | $1$ | $5$ | $3$ | $1$ |

$(1)$ Find the mode and median of the donation amounts of these $10$ students;<br/>$(2)$ Find the average donation amount of these $10$ students;<br/>$(3)$ If the school has a total of $2200$ students, estimate the total donation amount of the school based on the average donation amount of these $10$ students.

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