
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 store sells a batch of football souvenir books, with a cost price of $40$ yuan per book and a selling price of $44$ yuan per book. The store can sell 300 books per day. The store decides to increase the selling price, and after investigation, it is found that for every $1$ yuan increase in price, the daily sales decrease by 10 books. Let the new selling price after the increase be $x$ yuan $\left(44\leqslant x\leqslant 52\right)$, and let the daily sales be $y$ books.
$(1)$ Write down the functional relationship between $y$ and $x$;
$(2)$ At what price per book will the store maximize the profit from selling the souvenir books each day, and what is the maximum profit in yuan?

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