
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
"The passion of ice and snow, create the future", the Beijing Winter Olympics will arrive as scheduled on February 4, 2022. A certain merchant has already started selling souvenirs of the Winter Olympics mascot "Bing Dwen Dwen". The cost price of each souvenir is 40 yuan, and the selling price is set not lower than 44 yuan and not higher than 52 yuan. During the sales period, it was found that when the selling price was set at 44 yuan, 300 items could be sold per day. For every 1 yuan increase in selling price, the daily sales volume decreased by 10 items. Now the merchant decides to increase the selling price for profit. Let the daily sales volume be y items and the selling price be x yuan.<br/>$(1)$ Write down the functional relationship between y and x and the range of values for the independent variable x;<br/>$(2)$ Find the selling price per souvenir when the merchant makes a daily profit of 2400 yuan;<br/>$(3)$ Determine at what selling price per souvenir the merchant maximizes the daily profit w yuan. What is the maximum profit?

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