
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 shopping mall sells desk lamps at a price of $40$ yuan, which were purchased for $30$ yuan each. On average, $600$ lamps are sold each month. A survey shows that for every $1$ yuan increase in the selling price, the sales volume decreases by $10$ units.<br/>$(1)$ In order to achieve an average monthly profit of $10,000$ yuan, what should be the selling price of these desk lamps?<br/>$(2)$ After implementing a price increase, can the mall make a daily profit of $15,000$ yuan? Why?<br/>$(3)$ At what price should the desk lamps be sold to maximize profit, and what is the maximum profit?

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