
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
During the Qatar World Cup football match, a store sells a batch of football commemorative books, each costing $40. The selling price is set not to be less than $45, and the profit should not exceed $50\%$. It was found during the sales period that when the selling price is set at $45, 310 books can be sold per day. For every $1 increase in the selling price, the daily sales volume decreases by 10 books. Now the store decides to increase the price for sales. Let $y$ represent the daily sales and $x$ represent the selling price of each book.  
$(1)$ Write down the function relationship between $y$ and $x$ directly and the range of the independent variable $x$.  
$(2)$ At what price should the selling price of the football commemorative book be set so that the store can maximize the profit $w$ earned from selling books each day? What is the maximum profit?  
$(3)$ At what price per book will the store make a profit of $2600$ per day?

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