
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 books, with a cost of $20$ yuan per book. For marketing reasons, it is required that the selling price of each commemorative book should not be less than $20$ yuan and not more than $28$ yuan. It was found during the sales process that the weekly sales volume $y$ (books) of the commemorative book satisfies a linear function relationship with the selling price per book $x$ (yuan) given by $y=-2x+80$.

$(1)$ When the stationery store makes a profit of $150$ yuan from selling this type of commemorative book each week, what is the selling price per book?

$(2)$ Let the profit obtained by the stationery store from selling this type of commemorative book each week be $w$ yuan. At what selling price per book should the commemorative book be sold to maximize the profit obtained by the store? What is the maximum profit?

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