
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 stationery store purchased a batch of commemorative albums, with a cost of $20$ yuan per book. For marketing reasons, it is required that the selling price of each commemorative album is not less than $20$ yuan and not more than $28$ yuan. During the sales process, it was found that the weekly sales volume $y$ (in books) of the commemorative album satisfies a linear function relationship with the selling price per book $x$ (in yuan): when the selling price is $22$ yuan, the sales volume is $36$ books; when the selling price is $24$ yuan, the sales volume is $32$ books. <br/>$(1)$ Find the function relationship between $y$ and $x$; <br/>$(2)$ When the profit from selling these commemorative albums per week is $150$ yuan, what is the selling price per book? <br/>$(3)$ If the profit obtained by the stationery store from selling these commemorative albums per week is $w$ yuan, at what selling price per book will the store maximize the profit from selling these albums? What is the maximum profit?

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