
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 educational technology company sells two types of multimedia, $A$ and $B$, with the cost and selling price as shown in the table below:

|         | $A$  | $B$  |
|---------|------|------|
| Cost    | $3$  | $2.4$|
| Price   | $3.3$| $2.8$|

$(1)$ If the educational technology company plans to purchase a total of $50$ sets of these two types of multimedia, requiring a total of $132$ million yuan, how many sets of type $A$ and type $B$ multimedia does the company plan to purchase?

$(2)$ If the educational technology company plans to purchase a total of $50$ sets of these two types of multimedia, with $m$ sets of type $A$ multimedia purchased ($10 < m < 20$), and all purchased multimedia are sold, what is the maximum profit obtained by purchasing how many sets of type $A$ multimedia?

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