
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 city government strongly supports college students' entrepreneurship. Li Ming invested in selling a type of eye-protecting desk lamp with a cost price of $20$ yuan per unit under the government's support. During the sales process, it was found that if it is sold at a unit price of $26$ yuan, then $240$ lamps can be sold in one month. Based on sales experience, increasing the selling price will lead to a decrease in sales volume, that is, for every $1$ yuan increase in selling price, the sales volume decreases by $10$ units. According to the regulations of the price department, the selling price of this eye-protecting desk lamp must not be lower than the cost price and must not exceed $32$ yuan.  
$(1)$ Find the functional relationship between the monthly sales volume $y$ (units) and the selling price $x$ (yuan);  
$(2)$ Let Li Ming's monthly profit be $w$ (yuan). At what price should the selling price be set to maximize the monthly profit, and what is the maximum monthly profit?

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