
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. On average, $600$ lamps are sold each month. A survey shows that within the price range of $40$ to $60$ yuan, for every $1$ yuan increase in price, the sales volume decreases by $10$ lamps. The shopping mall decides to increase the price by $x$ yuan $(0 < x < 20)$.

$(1)$ After the price increase of $x$ yuan, the shopping mall can sell an average of ______ lamps per month (expressed as an algebraic expression in terms of $x$);  
$(2)$ In order to achieve an average monthly profit of $10000$ yuan, what should be the price of the desk lamp? How many lamps should be purchased at that time?  
$(3)$ At what price should the desk lamp be set to maximize the monthly sales profit?

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