
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 19th Asian Games was grandly held in Hangzhou. The government encourages the public to strengthen physical exercise. Li Ming invested in selling a type of table tennis racket with a cost price of $50 per piece with the support of the government. During the sales process, it was found that the relationship between the monthly sales volume $y$ (pieces) and the unit selling price $x$ (yuan) can be approximately regarded as a linear function: $y = -10x + 900$.
$(1)$ Let the monthly profit be $W$ (yuan). Find the expression of $W$ in terms of $x$.
$(2)$ At what price per unit should the selling price be set to maximize the monthly profit? What is the maximum monthly profit?
$(3)$ If the price department stipulates that the selling price of this type of table tennis racket must not exceed $75 yuan, and Li Ming wants to ensure a monthly profit of at least $3000 yuan, find the range of values for the selling price $x$.

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