
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 shopping mall is conducting a trial sale of a toy, with a cost price of $20$ yuan per piece. The mall has an agreement with the supplier that the profit during the trial sale period will not exceed $30\%$, and the selling price will remain the same within the same week. From the trial sale records, it is observed that when the selling price is $22$ yuan, 80 pieces of the toy were sold in a week; when the selling price is $24$ yuan, 60 pieces of the toy were sold in a week. The weekly sales volume $y$ (pieces) and the selling price $x$ (yuan) of the toy follow a linear function relationship.<br/>$(1)$ Find the relationship between the weekly sales volume $y$ (pieces) and the selling price $x$ (yuan);<br/>$(2)$ If the profit obtained from selling the toy in the mall in one week is $210$ yuan, what is the selling price of the toy?<br/>$(3)$ At what selling price will the mall maximize the profit from selling the toy in one week? What is the maximum profit $W$ in yuan?

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