
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 product has a cost price of $30$ yuan. During the sales period, it was found that when the selling price is set at $40$ yuan, 600 units can be sold per day. Due to the high demand, the merchant decided to increase the selling price. After market research, it was discovered that for every $1$ yuan increase in the selling price, the daily sales volume decreases by 10 units.<br/>$(1)$ When the selling price is $50$ yuan, the merchant earns ______ yuan per day;<br/>$(2)$ The price management department stipulates that the selling price of the product should not be less than $40$ yuan and not more than $60$ yuan. At what price should the selling price of the product be set so that the merchant maximizes the profit $w$ yuan per day? What is the maximum profit in yuan?

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