
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 Taobao shop sells desk lamps, with a cost of $30$ yuan each. Big data analysis shows that when the selling price of each desk lamp is $40$ yuan, an average of $600$ lamps are sold per month. If the price increases by $1$ yuan, the monthly sales decrease by $20 lamps, and if the price decreases by $1$ yuan, the monthly sales increase by $200 lamps. <br/>$(1)$ If the price decreases by $x$ yuan $\left(x \gt 0\right)$, the number of lamps sold per month is ______. <br/>$(2)$ To welcome "Double Eleven," the shop decides to offer a discount. With a stock of $1210$ desk lamps, and an expected monthly profit of exactly $8400$ yuan, find the selling price of each desk lamp.

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