
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 online store sells a certain children's clothing item for $80$ yuan each, and can sell $20$ items per day. In order to increase sales, increase profits, and reduce inventory quickly, the online store decides to sell at a discounted price. Market research shows that for every $1$ yuan reduction in price, they can sell an additional $2$ items per day. It is known that the cost price of each item of this children's clothing is $60$ yuan. Let $x$ represent the selling price of each item of this children's clothing, and $y$ represent the daily sales volume.

$(1)$ Find the functional relationship between $y$ and $x$;  
$(2)$ At what price per item will the online store make a profit of $432$ yuan per day?  
$(3)$ At what price per item will the online store maximize its daily sales profit?

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