
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 certain souvenir is very popular among tourists. The cost price of each souvenir is $40$ yuan, and the selling price is set to be no less than $44$ yuan and no more than $52 yuan. During the sales period, a merchant found that when the selling price is set at $44$ yuan, 300 souvenirs can be sold per day. For every increase of $1$ yuan in the selling price, the daily sales volume decreases by 10 units. Now the merchant decides to increase the selling price for better profit. Let the daily sales volume be $y$ units and the selling price be $x$ yuan. <br/>$(1)$ Write down the functional relationship between $y$ and $x$ and the range of the independent variable $x$; <br/>$(2)$ At what selling price should the souvenir be set so that the merchant can maximize the profit $w$ yuan per day? What is the maximum profit in yuan? <br/>$(3)$ The merchant donates $200$ yuan from the daily profit for charity. In order to ensure that the remaining profit after donation is not less than $2200$ yuan per day, find the range of the selling price $x$.

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