
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 FIFA World Cup in Russia, a certain store sells a batch of football commemorative books. The cost price of each book is $40$ yuan, and the selling price is set not less than $44$ yuan, with a profit margin not exceeding $30\%$. It was found during the sales period that when the selling price is set at $44$ yuan, 300 books can be sold per day. For every increase of $1$ yuan in the selling price, the daily sales decrease by 10 books. The store has decided to increase the selling price. Let $y$ represent the daily sales volume and $x$ represent the selling price.

$(1)$ 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 books be set for the store to maximize the profit $w$ yuan obtained from selling books each day? What is the maximum profit?

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