
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
Pengpeng Children's Clothing Store sells a certain type of children's clothing for $60$ yuan each, with a weekly sales volume of $100$ pieces. In order to promote sales, the store decides to reduce the price. According to market research, for every $1$ yuan reduction in price, an additional $10$ pieces can be sold per week. It is known that the cost of each piece of this children's clothing is $30$ yuan. Let $x$ represent the selling price per piece of this children's clothing and $y$ represent the weekly sales volume.

$(1)$ Find the functional relationship between $y$ and $x$ (without specifying the range of the independent variable).

$(2)$ When the selling price per piece is set to what value, the weekly sales profit is maximized, and what is the maximum profit?

$(3)$ When the selling price per piece is set to what value, the store can make a profit of $3910$ yuan per week?

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