
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 Winter Olympics and Paralympics held in China in 2022 have attracted worldwide attention. The mascots Bing Dwen Dwen and Shuey Rhon Rhon have become best-selling products. A certain merchant purchased a batch of Bing Dwen Dwen and Shuey Rhon Rhon sets at a price of $34$ yuan per set. If the selling price of each set is $48$ yuan, 200 sets can be sold per day. If the selling price is increased by $2$ yuan per set, 4 fewer sets are sold per day.  
$(1)$ If the selling price of each Bing Dwen Dwen and Shuey Rhon Rhon set is set at $x$ yuan, find the functional relationship between the sales volume $y$ and $x$.  
$(2)$ Find at what price per set the profit $W$ from selling the sets per day is maximized, and what is the maximum profit?

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