
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 clothing supermarket purchased several pieces of children's clothing at a unit price of $30$ yuan. The pricing department stipulates that the selling price should not be less than $30$ yuan per item, nor exceed $60$ yuan per item. After a period of sales, it was found that when the selling price was $60$ yuan, the average monthly sales volume was 80 pieces. However, when the selling price was reduced by $10$ yuan, an additional 20 pieces could be sold on average each month. Additionally, during the sales process, other expenses of $450$ yuan had to be paid monthly. Let the selling price be $x$ yuan and the average monthly sales volume be $y$ pieces.<br/>$(1)$ Find the functional relationship between $y$ and $x$.<br/>$(2)$ At what price per unit will the profit from selling this type of children's clothing be maximized each month? What is the maximum profit?

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