
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 is conducting a trial sale of a type of clothing with a cost of $60 per item. It is stipulated that during the trial sale period, the selling price must not be lower than the cost price, and the profit must not exceed 40%. It was found during the trial sale that the sales volume $y$ (in units) and the selling price $x$ (in dollars) follow a linear function $y=-x+120$.
- $(1)$ When the selling price is $80$ dollars, find the profit obtained by the shopping mall.
- $(2)$ If the shopping mall's profit is $W$ dollars, write down the relationship between profit $W$ and selling price $x$. At what selling price should the shopping mall set to maximize profit, and what is the maximum profit?
- $(3)$ If the shopping mall's profit is not less than $500, determine the range of selling price $x$.

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