
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
In recent years, China has been focusing on promoting educational equity, improving educational quality, accelerating the modernization of education, building a strong education country, and providing satisfactory education for the people. The work of digitalizing education continues to make progress, yielding fruitful results. In the process of digitalizing education, the role of multimedia cannot be underestimated. A certain educational technology company sells two types of multimedia teaching equipment, $A$ and $B$, with the cost and selling price of these two multimedia devices as shown in the table below. The educational technology company plans to purchase a total of 50 sets of $A$ and $B$ multimedia devices. Let $x$ represent the number of sets of type $A$ multimedia devices purchased, and $y$ represent the profit in million yuan.

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

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ If the company requires the quantity of type $B$ multimedia devices purchased not to exceed 4 times the quantity of type $A$ multimedia devices purchased, and when the company sells all the purchased two types of multimedia devices, find how many sets of type $A$ multimedia devices should be purchased to obtain the maximum profit, and what is the maximum profit in million yuan?

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