
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
In the era of "Internet $+$", online shopping is favored by consumers. A certain online store specializes in selling a pair of casual pants, with a cost of $40$ yuan per pair. When the selling price is $80$ yuan per pair, they can sell $100$ pairs per month. In order to attract more customers, the online store has taken a price reduction measure. According to market research, for each yuan reduction in the price of each pair of pants, they can sell an additional $5$ pairs per month. Let the selling price of each pair of pants be $x$ yuan $(x$ is a positive integer$)$, and the monthly sales volume be $y$ pairs.<br/>$(1)$ Find the functional relationship between $y$ and $x$ (do not need to write the range of the independent variable);<br/>$(2)$ Let the monthly profit obtained by the online store be $w$ yuan. By how much should the price of each pair of pants be reduced for the online store to maximize the monthly profit, and what is the maximum profit?

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