
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 a certain school, ninth-grade students Xiao Li, Xiao Qiang, and Xiao Hong participated in a social practice activity at a shopping mall. During the activity, they were involved in the sales of a certain product. It is known that the cost price of the product is $40$ yuan per item, and the selling price is $60$ yuan per item. Below is their conversation after the activity:

Xiao Li: I found that if this product is sold at $60$ yuan per item, we can sell $300$ items per week.

Xiao Qiang: I found that by adjusting the price based on the selling price of $60$ yuan per item, for every increase of $1$ yuan, we sell $10$ items less per week compared to the sales volume investigated by Xiao Li.

Xiao Hong: I found that by adjusting the price based on the selling price of $60$ yuan per item, for every decrease of $1$ yuan, we sell $20$ items more per week compared to the sales volume investigated by Xiao Li.

$(1)$ If the price is increased by $x$ yuan per item, then the actual number of items sold per week is ______, and the relationship between the profit $y_{1}$ (in yuan) from selling the product per week and $x$ is $y_{1}=$______, with the range of values for $x$ being ______;

$(2)$ If the price is decreased by $a$ yuan per item, then the relationship between the profit $y_{2}$ (in yuan) from selling the product per week and $a$ is $y_{2}=$______;

$(3)$ Under the condition of price increase, how should the pricing be set to maximize the profit from selling the product per week? What is the maximum profit?

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